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Evokation
 
 
Index
 

 

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
=
7
=
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
=
7
=
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
=
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
=
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
=
9
=
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
=
9
=
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

-
THE SOLAR SYSTEM
-
-
-
3
THE
33
15
6
5
SOLAR
65
29
2
6
SYSTEM
101
38
2
14
THE SOLAR SYSTEM
199
82
10
1+4
-
1+9+9
8+2
1+0
5
THE SOLAR SYSTEM
19
10
1
-
-
1+9
1+0
-
5
THE SOLAR SYSTEM
10
1
1
-
-
1+0
-
-
5
THE SOLAR SYSTEM
1
1
1

 

 

3
SUN
54
9
9
7
MERCURY
103
40
4
5
VENUS
81
18
9
5
EARTH
52
25
7
4
MOON
57
21
3
4
MARS
51
15
6
7
JUPITER
99
36
9
6
SATURN
93
21
3
6
URANUS
94
22
4
7
NEPTUNE
95
32
5
5
PLUTO
84
21
3
59
First Total
863
260
62
5+9
Add to Reduce
8+6+3
2+6
6+2
16
Second Total
17
8
8
1+6
Reduce to Deduce
1+7
-
-
7
Essence of Number
8
8
8

 

 

3
SUN
54
9
9
5
EARTH
52
25
7
4
MOON
57
21
3
12
First Total
163
55
19
1+2
Add to Reduce
1+6+3
5+5
6+2
3
Second Total
10
10
10
-
Reduce to Deduce
1+0
1+0
1+0
3
Essence of Number
7
7
7

 

 

L
=
3
-
4
LOOK
53
17
8
A
=
1
-
6
ABROAD
41
23
5
T
=
2
-
7
THROUGH
97
43
7
N
=
5
-
7
NATURES
98
26
8
R
=
9
-
5
RANGE
45
27
9
N
=
5
-
7
NATURES
98
26
8
M
=
4
-
6
MIGHTY
82
37
1
L
=
3
-
3
LAW
36
9
9
I
=
9
-
2
IS
28
10
1
C
=
3
-
6
CHANGE
38
29
2
-
-
44
4
53
First Total
616
247
58
-
-
4+4
-
5+3
Add to Reduce
6+1+6
2+4+7
5+8
Q
-
8
-
8
Second Total
13
13
13
-
-
-
-
-
Reduce to Deduce
1+3
1+3
1+3
-
-
8
-
8
Essence of Number
4
4
4

 

 

F
=
6
-
3
FOR
39
21
3
E
=
5
-
5
EVERY
75
30
3
A
=
1
-
6
ACTION
62
26
8
T
=
2
-
5
THERE
56
29
2
I
=
9
-
2
IS
28
19
1
A
=
1
-
2
AN
15
6
6
E
=
5
-
5
EQUAL
56
20
2
A
=
1
-
3
AND
19
10
1
O
=
6
-
8
OPPOSITE
115
43
7
R
=
9
-
8
REACTION
85
40
4
-
-
45
4
46
First Total
550
244
37
-
-
4+5
-
4+6
Add to Reduce
5+5+0
2+4+4
3+7
Q
-
9
-
10
Second Total
10
10
10
-
-
-
4
1+0
Reduce to Deduce
1+0
1+0
1+0
-
-
9
5
1
Essence of Number
1
1
1

 

 

S
=
1
-
5
SPEED
49
31
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
12
-
12
Add to Reduce
126
72
9
-
-
1+2
-
1+2
Reduce to Deduce
1+2+6
7+2
3+3
Q
-
3
-
3
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
5
SPEED
49
31
4
-
-
-
-
-
-
-
-
-
-
O
=
6
-
2
OF
21
12
3
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
-
-
10
-
12
Add to Reduce
126
72
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SPEED
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
P
=
7
3
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
34
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
27
-
5
SPEED
49
31
22
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
OF
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
16
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
F
=
6
17
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
-
-
38
-
2
OF
21
12
12
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
LIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
8
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
23
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
19
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
44
-
5
LIGHT
56
29
29
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SPEED
49
31
4
-
-
-
-
-
-
-
-
-
-
O
=
6
-
2
OF
21
12
3
-
1
2
3
4
10
12
14
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
1+0
1+2
1+4
-
-
-
-
10
-
12
Add to Reduce
126
72
9
-
1
2
3
4
1
3
5
8
9
-
-
1+0
-
1+2
Reduce to Deduce
1+2+6
7+2
3+3
-
-
-
-
-
-
-
-
-
-
Q
-
1
-
3
Essence of Number
9
9
9
-
1
2
3
4
1
3
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
5
SPEED
49
31
4
-
-
-
-
-
-
-
-
-
-
O
=
6
-
2
OF
21
12
3
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
-
-
10
-
12
Add to Reduce
126
72
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
P
=
7
3
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
34
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
O
=
6
16
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
F
=
6
17
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
=
3
8
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
23
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
19
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SPEED
49
31
4
-
-
-
-
-
-
-
-
-
-
O
=
6
-
2
OF
21
12
3
-
1
2
3
4
10
12
14
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
1+0
1+2
1+4
-
-
-
-
10
-
12
Add to Reduce
126
72
9
-
1
2
3
4
1
3
5
8
9
-
-
1+0
-
1+2
Reduce to Deduce
1+2+6
7+2
3+3
-
-
-
-
-
-
-
-
-
-
Q
-
1
-
3
Essence of Number
9
9
9
-
1
2
3
4
1
3
5
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
5
SPEED
49
31
4
-
-
-
-
-
-
-
-
-
-
O
=
6
-
2
OF
21
12
3
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
-
-
10
-
12
Add to Reduce
126
72
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
8
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
34
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
16
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
F
=
6
17
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
P
=
7
3
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
19
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
9
23
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SPEED
49
31
4
-
-
-
-
-
-
-
-
-
-
O
=
6
-
2
OF
21
12
3
-
1
2
3
4
10
12
14
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
1+0
1+2
1+4
-
-
-
-
10
-
12
Add to Reduce
126
72
9
-
1
2
3
4
1
3
5
8
9
-
-
1+0
-
1+2
Reduce to Deduce
1+2+6
7+2
3+3
-
-
-
-
-
-
-
-
-
-
Q
-
1
-
3
Essence of Number
9
9
9
-
1
2
3
4
1
3
5
8
9

 

 

7
MORPHIC
82
46
1
9
RESONANCE
94
40
4
16
First Total
176
86
5
1+6
Add to Reduce
1+7+6
8+6
-
7
Second Total
14
14
5
-
Reduce to Deduce
1+4
1+4
-
7
Essence of Number
5
5
5

 

 

4
MIND
40
22
4
6
MATTER
77
23
5
10
Add to Reduce
117
45
9
1+0
Reduce to Deduce
1+1+7
4+5
-
1
Essence of Number
9
9
9

 

 

-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
-
9
14
-
-
-
-
-
-
-
-
+
=
23
2+3
=
5
=
5
=
5
-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
+
=
31
3+1
=
4
=
4
=
4
-
-
13
-
-
4
-
13
1
20
20
5
18
+
=
94
9+4
=
13
1+3
4
=
4
-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
+
=
117
1+2
=
9
=
9
=
9
-
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
-
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-`
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
2
=
4
=
4
3
-
--
--
--
--
--
--
--
--
--
--
--
-
-
3
-
-
-
-
-
-
-
-
-`
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
--
--
--
--
--
--
--
--
--
--
-
-
6
--
--
--
--
--
--
--
7
-
-
--
--
--
--
--
--
--
--
--
--
-
-
7
--
--
--
--
--
--
--
8
-
-
--
--
--
--
--
--
--
--
--
--
-
-
8
--
--
--
--
--
--
--
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
24
10
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
10
-
45
-
18
2+4
1+0
-
9
-
-
-
-
-
-
-
-
9
-
-
2+1
-
-
1+0
-
4+5
-
1+8
6
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9
-
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
6
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9

 

 

10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
9
14
-
-
-
-
-
-
-
-
+
=
23
2+3
=
5
=
5
=
5
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
+
=
31
3+1
=
4
=
4
=
4
-
13
-
-
4
-
13
1
20
20
5
18
+
=
94
9+4
=
13
1+3
4
=
4
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
+
=
117
1+2
=
9
=
9
=
9
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
10
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
2
=
4
=
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
10
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
10
-
45
-
18
1+0
-
9
-
-
-
-
-
-
-
-
9
-
-
2+1
-
-
1+0
-
4+5
-
1+8
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9
-
4
9
5
4
-
4
1
2
2
5
9
+
=
45
1+0
=
9
=
9
=
9
1
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
1
-
9
-
9

 

 

-
MIND MATTER
-
-
-
1
M
13
4
4
1
I
9
9
9
2
ND
18
9
9
2
MA
14
5
5
3
TTE
45
9
9
1
R
18
9
9
10
MATTER MIND
117
45
45
1+0
1+1+7
4+5
4+5
1
MIND MATTER
9
9
9

 

 

-
-
-
-
-
MATTER ANTI MATTER
-
-
-
M
=
4
-
3
MATTER
77
23
5
A
=
1
-
7
ANTI
44
17
8
M
=
4
--
5
MATTER
77
23
5
-
-
9
-
15
MATTER ANTI MATTER
198
63
18
-
-
-
-
1+5
-
1+9+8
6+3
1+8
--
--
9
--
6
MATTER ANTI MATTER
18
9
9
-
-
-
--
-
-
1+8
-
-
-
-
9
--
6
MATTER ANTI MATTER
9
9
9

 

 

-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
-
-
-
-
-
-
-
+
=
14
1+4
=
5
-
5
-
5
-
-
-
-
-
-
-
-
-
-
14
-
9
-
-
-
-
-
-
-
+
=
23
2+3
=
5
-
5
-
5
-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
4
1
2
2
5
9
-
1
-
2
-
-
4
1
2
2
5
9
+
=
49
4+9
=
13
1+3
4
-
4
-
-
13
1
20
20
5
18
-
1
-
20
-
-
13
1
20
20
5
18
+
=
175
1+7+5
=
13
1+3
4
-
4
-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
13
1
20
20
5
18
-
1
14
20
9
-
13
1
20
20
5
18
+
=
198
1+9+8
=
18
1+8
9
9
-
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
+
=
63
6+3
=
9
-
9
9
-
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-`
-
1
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
3
=
3
-
3
-
-
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
-
-
-
2
occurs
x
5
=
10
1+0
1
3
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
3
-
-
-
-
-
-
-
-
-`
4
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
-
8
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
6
-
-
-
-
-
-
-
7
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
7
-
-
-
-
-
-
-
8
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
24
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
21
-
-
20
-
63
-
27
2+4
1+6
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
9
-
-
2+1
-
-
2+0
-
6+3
-
2+7
6
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
6
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
-
-
-
-
-
-
-
+
=
14
1+4
=
5
-
5
-
5
-
-
-
-
-
-
-
-
-
14
-
9
-
-
-
-
-
-
-
+
=
23
2+3
=
5
-
5
-
5
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
4
1
2
2
5
9
-
1
-
2
-
-
4
1
2
2
5
9
+
=
49
4+9
=
13
1+3
4
-
4
-
13
1
20
20
5
18
-
1
-
20
-
-
13
1
20
20
5
18
+
=
175
1+7+5
=
13
1+3
4
-
4
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
13
1
20
20
5
18
-
1
14
20
9
-
13
1
20
20
5
18
+
=
198
1+9+8
=
18
1+8
9
9
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
+
=
63
6+3
=
9
-
9
9
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
3
=
3
-
3
-
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
-
-
-
2
occurs
x
5
=
10
1+0
1
-
4
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
2
=
8
-
8
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
16
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
21
-
-
20
-
63
-
27
1+6
-
-
-
2
2
-
-
-
-
-
2
-
-
-
-
2
2
-
-
-
2+1
-
-
2+0
-
6+3
-
2+7
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
4
1
2
2
5
9
-
1
5
2
9
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
7
M
A
T
T
E
R
-
A
N
T
I
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

M
=
4
4
MIND
40
22
4
M
=
4
6
MATTER
77
23
5
M
=
4
6
MATTER
77
23
5
M
=
4
4
MIND
40
22
4
-
-
16
20
Add to Reduce
234
90
18
-
-
1+6
2+0
Reduce to Deduce
2+3+4
9+0
1+8
-
-
7
2
Essence of Number
9
9
9

 

 

-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
9
14
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
14
-
+
=
46
4+6
=
10
1+0
1
=
1
-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
-
-
4
+
=
62
6+2
=
8
=
8
=
8
-
-
13
-
-
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
-
-
4
+
=
188
1+8+8
=
17
1+7
8
-
8
-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
9
14
4
+
=
234
2+3+4
=
9
=
9
=
9
-
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
+
=
90
9+0
=
9
=
9
=
9
-
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
-
-
-
-
-
-
-`
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
-
2
2
-
-
-
-
-
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
3
-
-
-
-
-
-
-
-
-`
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
-
--
--
--
--
--
6
-
-
-
-
-
-
-
7
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
-
--
--
--
--
--
7
-
-
-
-
-
-
-
8
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
-
--
--
--
--
--
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
24
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
21
-
-
20
-
90
-
36
2+4
2+0
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
-
-
2+0
-
9+0
-
3+6
6
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9
-
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
-
-
-
-
-
-
-
-
-
-
6
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9

 

 

0
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
-
9
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
9
14
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
14
-
+
=
46
4+6
=
10
1+0
1
=
1
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
4
-
-
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
-
-
4
+
=
62
6+2
=
8
=
8
=
8
-
13
-
-
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
-
-
4
+
=
188
1+8+8
=
17
1+7
8
-
8
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
--
-
-
-
-
-
13
9
14
4
-
13
1
20
20
5
18
-
13
1
20
20
5
18
-
13
9
14
4
+
=
234
2+3+4
=
9
=
9
=
9
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
+
=
90
9+0
=
9
=
9
=
9
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
2
2
-
-
-
-
-
2
2
-
-
-
-
-
-
-
-
-
2
occurs
x
4
=
8
=
8
-
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
20
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
21
-
-
20
-
90
-
36
2+0
4
-
-
4
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
4
-
-
4
-
-
2+0
-
9+0
-
3+6
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9
-
4
9
5
4
-
4
1
2
2
5
9
-
4
1
2
2
5
9
-
4
9
5
4
-
-
-
-
-
-
-
-
-
-
2
M
I
N
D
-
M
A
T
T
E
R
-
M
A
T
T
E
R
-
M
I
N
D
-
-
3
-
-
2
-
9
-
9

 

 

-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
-
-
-
-
-
9
14
-
-
-
9
14
-
-
-
-
-
-
-
-
+
=
46
4+6
=
10
1+0
1
=
1
-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
4
1
2
2
5
9
-
4
-
-
4
-
4
-
-
4
-
4
1
2
2
5
9
+
=
62
6+2
=
8
=
8
=
8
-
-
13
1
20
20
5
18
-
13
-
-
4
-
13
-
-
4
-
13
1
20
20
5
18
+
=
188
1+8+8
=
17
1+7
8
=
8
-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
13
1
20
20
5
18
-
13
9
14
4
-
13
9
14
4
-
13
1
20
20
5
18
+
=
234
2+3+4
=
9
=
9
=
9
-
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
+
=
90
9+0
=
9
=
9
=
9
-
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-`
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
3
-
-
-
-
-
-
-
-
-`
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
6
-
-
-
-
-
-
-
7
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
7
-
-
-
-
-
-
-
8
-
--
--
--
--
--
--
--
-
--
--
--
--
-
--
--
--
--
--
--
--
--
--
--
--
--
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
4
=
36
3+6
9
24
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
20
-
90
-
36
2+4
2+0
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
2+1
-
-
2+0
-
9+0
-
3+6
6
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
6
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
-
-
9
5
-
-
-
-
-
-
-
-
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
-
-
-
-
9
14
-
-
-
9
14
-
-
-
-
-
-
-
-
+
=
46
4+6
=
10
1+0
1
=
1
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
4
1
2
2
5
9
-
4
-
-
4
-
4
-
-
4
-
4
1
2
2
5
9
+
=
62
6+2
=
8
=
8
=
8
-
13
1
20
20
5
18
-
13
-
-
4
-
13
-
-
4
-
13
1
20
20
5
18
+
=
188
1+8+8
=
17
1+7
8
=
8
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
--
-
-
-
-
-
13
1
20
20
5
18
-
13
9
14
4
-
13
9
14
4
-
13
1
20
20
5
18
+
=
234
2+3+4
=
9
=
9
=
9
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
+
=
90
9+0
=
9
=
9
=
9
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
-
-
-
-
-
-
-
-
-`
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
-
-
-
-
2
occurs
x
4
=
8
=
8
`
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
4
occurs
x
6
=
24
2+4
6
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
4
=
36
3+6
9
20
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
21
-
-
20
-
90
-
36
2+0
4
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
-
4
-
-
-
-
-
-
-
2+1
-
-
2+0
-
9+0
-
3+6
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9
-
4
1
2
2
5
9
-
4
9
5
4
-
4
9
5
4
-
4
1
2
2
5
9
-
-
-
-
-
-
-
-
-
-
2
M
A
T
T
E
R
-
M
I
N
D
-
M
I
N
D
-
M
A
T
T
E
R
-
-
3
-
-
2
-
9
-
9

 

 

-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
--
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
9
5
3
1
2
-
4
-
2
7
+
=
33
3+3
=
6
=
6
=
6
-
-
18
5
12
1
20
-
22
-
20
25
+
=
123
1+2+3
=
6
=
6
=
6
-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
18
5
12
1
20
9
22
9
20
25
+
=
141
1+4+1
=
6
=
6
=
6
-
-
9
5
3
1
2
9
4
9
2
7
+
=
51
5+1
=
6
=
6
=
6
-
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
--
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
3
-
-
-
--
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
7
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
14
10
R
E
L
A
T
I
V
I
T
Y
-
-
31
-``
-
10
-
51
-
33
1+4
1+0
9
-
-
-
-
9
-
9
-
-
-
-
3+1
-
-
1+0
-
5+1
-
3+3
5
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-``
-
1
-
6
-
6
-
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
-
-
-
-
-
5
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-
-
1
-
6
-
6

 

 

10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
+
=
18
1+8
=
9
=
9
=
9
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
9
5
3
1
2
-
4
-
2
7
+
=
33
3+3
=
6
=
6
=
6
-
18
5
12
1
20
-
22
-
20
25
+
=
123
1+2+3
=
6
=
6
=
6
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
18
5
12
1
20
9
22
9
20
25
+
=
141
1+4+1
=
6
=
6
=
6
-
9
5
3
1
2
9
4
9
2
7
+
=
51
5+1
=
6
=
6
=
6
10
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
3
-
-
-
--
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
--
-
-
7
-
-
7
occurs
x
1
=
7
-
7
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
10
R
E
L
A
T
I
V
I
T
Y
-
-
31
-``
-
10
-
51
-
33
1+0
9
-
-
-
-
9
-
9
-
-
-
-
3+1
-
-
1+0
-
5+1
-
3+3
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-``
-
1
-
6
-
6
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
-
-
-
-
-
1
R
E
L
A
T
I
V
I
T
Y
-
-
4
-
-
1
-
6
-
6

 

 

-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
8
-
6
-
-
-
6
-
-
-
-
-
-
-
9
-
9
-
-
+
=
43
4+3
=
7
=
7
=
7
-
-
-
-
14
-
-
-
-
-
-
8
-
15
-
-
-
15
-
-
-
-
-
-
-
9
-
9
-
-
+
=
70
7+0
=
7
=
7
=
7
-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
7
5
-
5
9
1
3
-
2
-
5
-
9
7
-
-
6
-
9
5
3
1
2
-
4
-
2
7
+
=
92
9+2
=
11
1+1
2
=
2
-
-
7
5
-
5
18
1
12
-
20
-
5
-
18
25
-
-
6
-
18
5
12
1
20
-
22
-
20
25
+
=
245
2+4+5
=
11
1+1
2
=
1
-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
7
5
14
5
18
1
12
-
20
8
5
15
18
25
-
15
6
-
18
5
12
1
20
9
22
9
20
25
+
+
351
1+8+2
=
9
-
9
=
9
-
-
7
5
5
5
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
+
+
135
1+3+5
=
9
-
9
=
2
-
25
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
3
=
6
=
6
3
``-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
5
5
5
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
7
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
5
=
45
=
9
45
25
G
E
N
E
R
A
L
T
H
E
O
R
Y
O
F
R
E
L
A
T
I
V
I
T
Y
-
-
45
-
-
25
-
135
-
54
4+5
2+5
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
4+5
-
-
2+5
-
1+3+5
-
5+4
9
7
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
7
-
9
-
9
-
-
7
5
5
5
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
-
-
-
-
-
9
7
G
E
N
E
R
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
7
-
9
-
9

 

 

G
=
7
7
GENERAL
62
35
8
T
=
2
6
THEORY
91
37
1
O
=
6
2
OF
21
12
3
R
=
9
10
RELATIVITY
141
51
6
-
-
24
25
Add to Reduce
315
135
18
-
-
2+4
2+5
Reduce to Deduce
3+1+5
1+3+5
1+8
-
-
6
7
Essence of Number
9
9
9

 

 

7
SPECIAL
65
29
2
6
THEORY
91
37
1
2
OF
21
12
3
10
RELATIVITY
141
51
6
25
First Total
318
129
12
2+5
Add to Reduce
3+1+8
1+2+9
1+2
7
Second Total
12
12
3
-
Reduce to Deduce
1+2
1+2
-
7
Essence of Number
3
3
3

 

 

G
=
7
7
GENERAL
62
35
8
T
=
2
6
THEORY
91
37
1
-
-
9
13
Add to Reduce
153
72
9
-
-
-
1+3
Reduce to Deduce
1+5+3
7+2
-
-
-
9
4
Essence of Number
9
9
9

 

 

T
=
2
3
THE
33
15
6
G
=
7
7
GENERAL
62
35
8
T
=
2
6
THEORY
91
37
1
O
=
6
2
OF
21
12
3
R
=
9
10
RELATIVITY
141
51
6
-
-
26
28
First Total
348
150
24
-
-
2+6
2+8
Add to Reduce
3+4+8
1+5+0
2+4
-
-
8
10
Second Total
15
6
6
-
-
-
1+0
Reduce to Deduce
1+5
-
-
-
-
8
1
Essence of Number
6
6
6

 

 

3
THE
33
15
6
7
SPECIAL
65
29
2
6
THEORY
91
37
1
2
OF
21
12
3
10
RELATIVITY
141
51
6
28
First Total
351
144
18
2+8
Add to Reduce
3+5+1
1+4+4
1+8
10
Second Total
9
9
9
1+0
Reduce to Deduce
-
-
-
1
Essence of Number
9
9
9

 

 

25
GENERAL THEORY OF RELATIVITY
315
135
9

 

 

28
THE SPECIAL THEORY OF RELATIVITY
351
144
9

 

 

-
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
1
1
-
-
-
9
-
-
-
-
8
-
6
-
-
-
6
-
-
-
-
-
-
-
9
-
9
-
-
+
=
56
5+6
=
11
1+1
2
=
2
-
-
-
8
-
-
19
-
-
-
9
-
-
-
-
8
-
15
-
-
-
15
-
-
-
-
-
-
-
9
-
9
-
-
+
=
92
9+2
=
11
1+1
2
=
2
-
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
-
7
5
3
-
1
3
-
2
-
5
-
9
7
-
-
6
-
9
5
3
1
2
-
4
-
2
7
+
=
88
8+8
=
16
1+6
7
=
7
-
-
20
-
5
-
-
16
5
3
-
1
12
-
20
-
5
-
18
25
-
-
6
-
18
5
12
1
20
-
22
-
20
25
+
=
259
2+5+9
=
16
1+6
7
=
7
-
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
19
16
5
3
9
1
12
-
20
8
5
15
18
25
-
15
6
-
18
5
12
1
20
9
22
9
20
25
+
+
351
3+5+1
=
9
-
9
=
9
-
-
2
8
5
-
1
7
5
3
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
+
+
144
1+4+4
=
9
-
9
=
9
218
28
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
2
``-
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
-
-
-
2
occurs
x
4
=
8
=
8
3
``-
-
-
-
-
-
-
-
3
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
3
=
9
-
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
5
=
45
=
9
45
28
T
H
E
S
P
E
C
I
A
L
T
H
E
O
R
Y
O
F
R
E
L
A
T
I
V
I
T
Y
-
-
45
-
-
28
-
144
-
54
4+5
2+8
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
-
9
-
-
-
-
4+5
-
-
2+8
-
1+4+4
-
5+4
9
10
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
10
-
9
-
9
-
1+0
2
8
5
-
1
7
5
3
9
1
3
-
2
8
5
6
9
7
-
6
6
-
9
5
3
1
2
9
4
9
2
7
-
-
-
-
-
1+0
-
-
-
-
9
1
T
H
E
-
S
P
E
C
I
A
L
-
T
H
E
O
R
Y
-
O
F
-
R
E
L
A
T
I
V
I
T
Y
-
-
9
-
-
1
-
9
-
9

 

 

SCHRODINGERS

CAT

 

S
=
1
-
12
SCHRODINGERS
139
67
4
C
=
3
-
3
CAT
24
6
6
D
=
4
-
3
DID
17
17
8
C
=
3
-
10
CURIOUSITY
160
52
7
K
=
2
-
4
KILL
44
17
6
T
=
2
-
3
THE
33
15
6
C
=
3
-
3
CAT
24
6
6
-
-
18
4
38
First Total
441
180
19
-
-
1+8
-
3+8
Add to Reduce
4+4+1
1+8+0
6+2
-
-
9
-
11
Second Total
9
10
10
4
4
-
4
1+1
Reduce to Deduce
-
1+0
1+0
Q
-
9
5
2
Essence of Number
9
7
7

 

 

-
COSMOLOGICAL CONSTANT
-
-
-
8
C+O+S+M+O+L+O+G
99
36
9
1
I
9
9
9
3
C+A+L
16
7
7
2
C+O
18
9
9
4
N+S+T+A
54
9
9
2
N+T
34
7
7
20
COSMOLOGICAL CONSTANT
230
77
50
2+0
-
2+3+0
7+7
5+0
2
COSMOLOGICAL CONSTANT
5
14
5
-
-
-
1+4
-
2
COSMOLOGICAL CONSTANT
5
5
5

 

 

-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
-
-
-
-
-
-
-
-
-
-
-
6
1
-
6
-
6
-
9
-
-
-
-
-
6
5
1
-
-
5
-
+
=
45
4+5
=
9
-
9
-
9
-
--
-
15
19
-
15
-
15
-
9
-
-
-
-
-
15
14
19
-
-
14
-
+
=
135
1+3+5
=
9
-
9
-
9
-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
4
-
3
-
7
-
3
1
3
-
3
-
-
-
2
1
-
2
+
=
32
3+2
=
5
-
5
-
5
-
-
3
-
-
13
-
12
-
7
-
3
1
12
-
3
-
-
-
20
1
-
20
+
=
95
9+5
=
14
1+4
5
-
5
-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
--
-
-
-
-
-
-
-
-
-
-
-
3
15
19
13
15
12
15
7
9
3
1
12
-
3
15
14
19
20
1
14
20
+
=
230
2+3+0
=
5
-
5
-
5
-
-
3
6
1
4
6
3
6
7
9
3
1
3
-
3
6
5
1
2
1
5
2
+
=
77
7+7
=
14
1+4
5
-
5
-
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
--
-
-
-
-
-
-
-
-
-
-
-
-
1
1
1
-
-
-
-
-
-
1
-
-
-
-
-
1
-
1
-
-
-
-
1
occurs
x
4
=
4
-
4
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
-
-
2
occurs
x
2
=
4
-
4
-
-
3
-
-
-
-
3
-
-
-
3
-
3
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
5
=
15
1+5
6
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
6
-
-
6
-
6
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
8
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
37
-
-
20
-
77
-
41
-
-
3
-
-
-
-
3
-
-
-
3
-
3
-
3
-
-
-
-
-
-
-
-
-
3+7
-
-
2+0
-
7+7
-
4+1
8
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
10
-
-
2
-
14
-
5
-
-
3
6
1
4
6
3
6
7
9
3
1
3
-
3
6
5
1
2
1
5
2
-
-
1+0
-
-
-
-
1+4
-
-
8
20
C
O
S
M
O
L
O
G
I
C
A
L
-
C
O
N
S
T
A
N
T
-
-
1
-
-
2
-
5
-
5

 

 

24
SUPERNATURAL SUPERSTITION
351
126
9
25
GENERAL THEORY OF RELATIVITY
315
135
9
26
ABCDEFGHIJKLMNOPQRSTUVWXYZ
351
135
9

 

 

 

-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
35
6
A
T
U
M
-
-
10
-
-
4
-
10
3+5
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
8
6
A
T
U
M
-
-
1
-
-
4
-
2

 

 

6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
4
occurs
x
1
=
4
6
A
T
U
M
-
-
10
-
-
4
-
10
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
6
A
T
U
M
-
-
1
-
-
4
-
2

 

AUTUMN ATUM AUTUMN

 

-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
-
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
30
6
A
U
T
U
M
N
-
-
15
-
-
6
-
18
3+0
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
3
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
3
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

 

6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
A
U
T
U
M
N
-
-
15
-
-
6
-
18
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

 

6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
A
U
T
U
M
N
-
-
15
-
-
6
-
18
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

AUTUMN ATUM AUTUMN

QUANTUM ATUM QUANTUM

 

-
QUANTUM
-
-
-
4
.......A...TUM.
55
10
1
7
QUANTUM
107
26
8
4
.......A...TUM.
55
10
1
7
QUANTUM
107
26
8
3
.....QU...N.............
52
16
7
7
QUANTUM
107
26
8

 

 

-
QUANTUM
-
-
-
1
Q
17
8
8
3
UAN
35
9
9
3
TUM
54
9
9
7
QUANTUM
107
26
26
-
-
1+0+7
2+6
2+6
7
QUANTUM
8
8
8

 

 

-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
1+2
3
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
-
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
6
SIX
-
-
-
-
7
``-
-
-
-
-
-
-
-
-
-
7
SEVEN
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
9
-
-
-
-
-
-
-
-
-
-
9
NINE
-
-
-
-
22
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
2+2
-
-
-
-
-
-
-
-``
-
-
2+3
-
-
-
-
2+6
4
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
4
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

 

7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
1+2
3
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
-
-
-
-
-
-
-
-``
-
-
2+3
-
-
-
-
2+6
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

 

7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
1+2
3
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
-
-
-
1
-
2
3
4
-
-
2+3
-
-
-
-
2+6
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

RA ATUM ATUM RA

 

 

Q
=
8
-
7
QUANTUM
107
26
8
E
=
5
-
8
ENERGIES
82
46
1
-
-
13
-
27
First Total
189
72
9
-
-
1+3
-
2+7
Add to Reduce
1+8+9
7+2
-
Q
-
4
-
9
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
5
9
Essence of Number
9
9
9

 

 

-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
9
-
1
+
=
20
2+0
=
2
-
2
=
2
-
-
-
-
-
14
-
-
-
-
-
14
-
-
-
9
-
19
+
=
56
5+6
=
11
1+1
2
=
2
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
5
-
5
9
7
-
5
-
+
=
52
5+2
=
7
-
7
=
7
-
-
17
21
1
-
20
21
13
-
5
-
5
18
7
-
5
-
+
=
133
1+3+3
=
7
-
7
=
7
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
5
14
5
18
7
9
5
19
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
+
=
72
7+2
=
9
-
9
=
9
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
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QUANTUM THOUGHTS GODS THOUGHTS QUANTUM

 

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QUANTUM ATUM AMUN ATEN ATOM QUANTUM

 

 

Quantum - Wikipedia, the free encyclopedia
In physics, a quantum (plural: quanta) is an indivisible entity of a quantity that has the units as the Planck constant and is related to both energy and ...
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Quantum
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"Quanta" redirects here. For other uses, see Quantum (disambiguation).
This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (February 2008)
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In physics, a quantum (plural: quanta) is an indivisible entity of a quantity that has the units as the Planck constant and is related to both energy and momentum of elementary particles of matter (called fermions) and of photons and other bosons. The word comes from the Latin "quantus", for "how much." Behind this, one finds the fundamental notion that a physical property may be "quantized", referred to as "quantization". This means that the magnitude can take on only certain discrete numerical values, rather than any value, at least within a range. There is a related term of quantum number.

A photon is often referred to as a "light quantum". The energy of an electron bound to an atom (at rest) is said to be quantized, which results in the stability of atoms, and of matter in general. But these terms can be a little misleading, because what is quantized is this Planck's constant quantity whose units can be viewed as either energy multiplied by time or momentum multiplied by distance.

Usually referred to as quantum "mechanics", it is regarded by virtually every professional physicist as the most fundamental framework we have for understanding and describing nature at the infinitesimal level, for the very practical reason that it works. It is "in the nature of things", not a more or less arbitrary human preference.

Contents [hide]
1 Development of quantum theory
1.1 The quantum black-body radiation formula
2 Beyond electromagnetic radiation
2.1 The birth of quantum mechanics
3 See also
4 References

[edit] Development of quantum theory
Quantum theory, the branch of physics which is based on quantization, began in 1900 when Max Planck published his theory explaining the emission spectrum of black bodies. In that paper Planck used the Natural system of units he invented the previous year. The consequences of the differences between classical and quantum mechanics quickly became obvious. But it was not until 1926, by the work of Werner Heisenberg, Erwin Schrödinger, and others, that quantum mechanics became correctly formulated and understood mathematically. Despite tremendous experimental success, the philosophical interpretations of quantum theory are still widely debated.

Planck was reluctant to accept the new idea of quantization, as were many others. But, with no acceptable alternative, he continued to work with the idea, and found his efforts were well received. Eighteen years later, when he accepted the Nobel Prize in Physics for his contributions, he called it "a few weeks of the most strenuous work" of his life. During those few weeks, he even had to discard much of his own theoretical work from the preceding years. Quantization turned out to be the only way to describe the new and detailed experiments which were just then being performed. He did this practically overnight, openly reporting his change of mind to his scientific colleagues, in the October, November, and December meetings of the German Physical Society, in Berlin, where the black body work was being intensely discussed. In this way, careful experimentalists (including Friedrich Paschen, O.R. Lummer, Ernst Pringsheim, Heinrich Rubens, and F. Kurlbaum), and a reluctant theorist, ushered in a momentous scientific revolution.

[edit] The quantum black-body radiation formula
When a body is heated, it emits radiant heat, a form of electromagnetic radiation in the infrared region of the EM spectrum. All of this was well understood at the time, and of considerable practical importance. When the body becomes red-hot, the red wavelength parts start to become visible. This had been studied over the previous years, as the instruments were being developed. However, most of the heat radiation remains infrared, until the body becomes as hot as the surface of the Sun (about 6000 K or 5726 °C, where most of the light is white in color). This was not achievable in the laboratory at that time. What is more, measuring specific infrared wavelengths was only then becoming feasible, due to newly developed experimental techniques. Until then, most of the electromagnetic spectrum was not measurable, and therefore blackbody emission had not been mapped out in detail.

The quantum black-body radiation formula, being the very first piece of quantum mechanics, appeared Sunday evening October 7, 1900, in a so-called back-of-the-envelope calculation by Planck. It was based on a report by Rubens (visiting with his wife) on the very latest experimental findings in the infrared. Later that evening, Planck sent the formula on a postcard, which Rubens received the following morning. A couple of days later, he informed Planck that it worked perfectly. At first, it was just a fit to the data; only later did it turn out to enforce quantization.

This second step was only possible due to a certain amount of luck (or skill, even though Planck himself called it "a fortuitous guess at an interpolation formula"). It was during the course of polishing the mathematics of his formula that Planck stumbled upon the beginnings of Quantum Theory. Briefly stated, he had two mathematical expressions:

(i) from the previous work on the red parts of the spectrum, he had x;
(ii) now, from the new infrared data, he got x².
Combining these as x(a+x), he still has x, approximately, when x is much smaller than a (the red end of the spectrum); but now also x² (again approximately) when x is much larger than a (in the infrared). The formula for the energy E, in a single mode of radiation at frequency λ, and temperature T, can be written

This is (essentially) what is being compared with the experimental measurements. There are two parameters to determine from the data, written in the present form by the symbols used today: h is the new Planck's constant, and k is Boltzmann's constant. Both have now become fundamental in physics, but that was by no means the case at the time. The "elementary quantum of energy" is hλ. But such a unit does not normally exist, and is not required for quantization.

[edit] Beyond electromagnetic radiation
While quantization was first discovered in electromagnetic radiation, it describes a fundamental aspect of energy not just restricted to photons.[1]

[edit] The birth of quantum mechanics
From the experiments, Planck deduced the numerical values of h and k. Thus he could report, in the German Physical Society meeting on December 14, 1900, where quantization (of energy) was revealed for the first time, values of the Avogadro-Loschmidt number, the number of real molecules in a mole, and the unit of electrical charge, which were more accurate than those known until then. This event has been referred to as "the birth of quantum mechanics".

[edit] See also
Quantum mechanics
Quantum state
Quantum number
Quantum cryptography
Quantum electronics
Quantum computer
Quantum chromodynamics
Quantum entanglement
Quantum coherence
Quantum immortality
Quantum lithography
Quantum sensor
Quantum dot
Quantum channel
Magnetic flux quantum
Quantum cellular automata
Quantization
Subatomic particle
Elementary particle
Photon polarization

[edit] References
1.^ Real-World Quantum Effects Demonstrated February 11, 2005

 

Quantum mechanics - Wikipedia, the free encyclopedia
Quantum mechanics (QM) is a set of principles describing physical reality at the atomic level of matter (molecules and atoms) and the subatomic (electrons, ...
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Quantum mechanics
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Fig. 1: Probability densities corresponding to the wavefunctions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: n = 1, 2, 3, ...) and angular momentum (increasing across from left to right: s, p, d,...). Brighter areas correspond to higher probability density in a position measurement. Wavefunctions like these are directly comparable to Chladni's figures of acoustic modes of vibration in classical physics and are indeed modes of oscillation as well: they possess a sharp energy and thus a keen frequency. The angular momentum and energy are quantized, and only take on discrete values like those shown (as is the case for resonant frequencies in acoustics).Quantum mechanics (QM) is a set of principles describing physical reality at the atomic level of matter (molecules and atoms) and the subatomic (electrons, protons, and even smaller particles). These descriptions include the simultaneous wave-like and particle-like behavior of both matter[1] and radiation[2] ("wave–particle duality"). In the quantum mechanics of a subatomic particle, one can never specify its state, such as its simultaneous location and velocity, with complete certainty (this is called the Heisenberg uncertainty principle — see its formula in the box to the right).

Certain systems, however, do exhibit quantum mechanical effects on a larger scale; superfluidity (the frictionless flow of a liquid at temperatures near absolute zero) is one well-known example. Quantum theory also provides accurate descriptions for many previously unexplained phenomena such as black body radiation and the stability of electron orbits. It has also given insight into the workings of many different biological systems, including smell receptors and protein structures.[3]

Even so, classical physics often can be a good approximation to results otherwise obtained by quantum physics, typically in circumstances with large numbers of particles or large quantum numbers. (However, some open questions remain in the field of quantum chaos.)

Contents [hide]
1 Overview
2 History
3 Quantum mechanics and classical physics
4 Theory
4.1 Mathematical formulation
4.2 Interactions with other scientific theories
5 Example
6 Attempts at a unified field theory
7 Relativity and quantum mechanics
8 Applications
9 Philosophical consequences
10 See also
11 Notes
12 References
13 Further reading
14 External links

[edit] Overview
Main article: Introduction to quantum mechanics
The word quantum is Latin for "how great" or "how much."[4] In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). The discovery that waves have discrete energy packets (called quanta) that behave in a manner similar to particles led to the branch of physics that deals with atomic and subatomic systems which we today call quantum mechanics. It is the underlying mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational chemistry, quantum chemistry, particle physics, and nuclear physics. The foundations of quantum mechanics were established during the first half of the twentieth century by Werner Heisenberg, Max Planck, Louis de Broglie, Albert Einstein, Niels Bohr, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli, David Hilbert, and others.[5] Some fundamental aspects of the theory are still actively studied.[6]

Quantum mechanics is essential to understand the behavior of systems at atomic length scales and smaller. For example, if classical mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus, making stable atoms impossible. However, in the natural world the electrons normally remain in an uncertain, non-deterministic "smeared" (wave-particle wave function) orbital path around or "through" the nucleus, defying classical electromagnetism.[7]

Quantum mechanics was initially developed to provide a better explanation of the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom was developed as an explanation for the electron's staying in its orbital, which could not be explained by Newton's laws of motion and by Maxwell's laws of classical electromagnetism.[8]

In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space.[9] This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with arbitrary accuracy. For instance, electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. Contours of constant probability, often referred to as “clouds” may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate.[10]

The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. This led to a theory of unity between subatomic particles and electromagnetic waves called wave–particle duality in which particles and waves were neither one nor the other, but had certain properties of both. While quantum mechanics describes the world of the very small, it also is needed to explain certain “macroscopic quantum systems” such as superconductors and superfluids.[11]

Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (I) the quantization (discretization) of certain physical quantities, (II) wave-particle duality, (III) the uncertainty principle, and (IV) quantum entanglement. Each of these phenomena is described in detail in subsequent sections.[11]

[edit] History
This section may require cleanup to meet Wikipedia's quality standards. Please improve this section if you can. (August 2009)

Main article: History of quantum mechanics
The history of quantum mechanics[12] began essentially with the 1838 discovery of cathode rays by Michael Faraday, the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that any energy is radiated and absorbed in quantities divisible by discrete ‘energy elements’, E, such that each of these energy elements is proportional to the frequency ν with which they each individually radiate energy, as defined by the following formula:

where h is Planck's Action Constant. Planck insisted[13] that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself. However, this did not explain the photoelectric effect (1839), i.e. that shining light on certain materials can function to eject electrons from the material. In 1905, basing his work on Planck’s quantum hypothesis, Albert Einstein[14] postulated that light itself consists of individual quanta. These later came to be called photons (1926). From Einstein's simple postulation was born a flurry of debating, theorizing and testing, and thus, the entire field of quantum physics.

[edit] Quantum mechanics and classical physics
Predictions of quantum mechanics have been verified experimentally to a very high degree of accuracy. Thus, the current logic of correspondence principle between classical and quantum mechanics is that all objects obey laws of quantum mechanics, and classical mechanics is just a quantum mechanics of large systems (or a statistical quantum mechanics of a large collection of particles). Laws of classical mechanics thus follow from laws of quantum mechanics at the limit of large systems or large quantum numbers.[15] However, chaotic systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.

The main differences between classical and quantum theories have already been mentioned above in the remarks on the Einstein-Podolsky-Rosen paradox. Essentially the difference boils down to the statement that quantum mechanics is coherent (addition of amplitudes), whereas classical theories are incoherent (addition of intensities). Thus, such quantities as coherence lengths and coherence times come into play. For microscopic bodies the extension of the system is certainly much smaller than the coherence length; for macroscopic bodies one expects that it should be the other way round.[16]

This is in accordance with the following observations:

Many “macroscopic” properties of “classic” systems are direct consequences of quantum behavior of its parts. For example, stability of bulk matter (which consists of atoms and molecules which would quickly collapse under electric forces alone), rigidity of this matter, mechanical, thermal, chemical, optical and magnetic properties of this matter—they are all results of interaction of electric charges under the rules of quantum mechanics.[17]

While the seemingly exotic behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with extremely fast-moving or extremely tiny particles, the laws of classical “Newtonian” physics still remain accurate in predicting the behavior of surrounding (“large”) objects—of the order of the size of large molecules and bigger—at velocities much smaller than the velocity of light.[18]

[edit] Theory
There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg[19])[20] and wave mechanics (invented by Erwin Schrödinger[21]).

In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).[22]

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions using probability distributions; that is, the probability of obtaining possible outcomes from measuring an observable. Oftentimes these results are skewed by many causes, such as dense probability clouds[23] or quantum state nuclear attraction.[24][25] Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. Hence, uncertainty is involved in the value. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" can be roughly translated from German as inherent or as a characteristic[26]). In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle for its position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it only provides a range of probabilities of where that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstate).

For example, consider a free particle. In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as the properties of a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over space as a wave function. The position and momentum of the particle are observables. The Uncertainty Principle states that both the position and the momentum cannot simultaneously be measured with full precision at the same time. However, one can measure the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large (a Dirac delta) at a particular position x and zero everywhere else. If one performs a position measurement on such a wavefunction, the result x will be obtained with 100% probability (full certainty). This is called an eigenstate of position (mathematically more precise: a generalized position eigenstate (eigendistribution)). If the particle is in an eigenstate of position then its momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum then its position is completely unknown. [27] In an eigenstate of momentum having a plane wave form, it can be shown that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate.[28]

Usually, a system will not be in an eigenstate of the observable we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. This process is known as wavefunction collapse, a debatable process.[29] It involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result.[30] It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.[31]

Wave functions can change as time progresses. An equation known as the Schrödinger equation describes how wave functions change in time, a role similar to Newton's second law in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates.[32] Some wave functions produce probability distributions that are constant or independent of time, such as when in a stationary state of constant energy, time drops out of the absolute square of the wave function. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (Fig. 1). (Note that only the lowest angular momentum states, labeled s, are spherically symmetric).[33]

The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time.[34] During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random. A time-evolution simulation can be seen here.[1]

The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Interpretations of quantum mechanics have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.[35]

[edit] Mathematical formulation
Main article: Mathematical formulation of quantum mechanics
See also: Quantum logic
In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac[36] and John von Neumann[37], the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues.

The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution.

The inner product between two state vectors is a complex number known as a probability amplitude. During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator - which explains the choice of Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.

The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states.

It turns out that analytic solutions of Schrödinger's equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion and the hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos.

An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics.

[edit] Interactions with other scientific theories
The fundamental rules of quantum mechanics are very deep. They assert that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators, or if it even exists. These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or equivalently, larger quantum numbers. In other words, classic mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit.

Unsolved problems in physics: In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "superposition of states" and "wavefunction collapse", give rise to the reality we perceive?When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field rather than a fixed set of particles. The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.

The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory, by the physicists Carl Jamieson, Sheldon Glashow and Steven Weinberg.

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity.

[edit] Example
Main article: Particle in a box
The particle in a 1-dimensional potential energy box is the most simple example where restraints lead to the quantization of energy levels. The box is defined as zero potential energy inside a certain interval and infinite everywhere outside that interval. For the 1-dimensional case in the x direction, the time-independent Schrödinger equation can be written as:[38]

The general solutions are:

or

(by Euler's formula).
The presence of the walls of the box restricts the acceptable solutions of the wavefunction. At each wall:

Consider x = 0

sin 0 = 0, cos 0 = 1. To satisfy the cos term has to be removed. Hence D = 0.
Now consider:

at x = L,
If C = 0 then for all x. This would conflict with the Born interpretation
therefore sin kL = 0 must be satisfied, yielding the condition.

In this situation, n must be an integer showing the quantization of the energy levels.

[edit] Attempts at a unified field theory
Main article: Grand unified theory
As of 2009 the quest for unifying the fundamental forces through quantum mechanics is still ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is currently the most accurately tested physical theory, [39] has been successfully merged with the weak nuclear force into the electroweak force and work is currently being done to merge the electroweak and strong force into the electrostrong force. Current predictions state that at around 1014 GeV the three aforementioned forces are fused into a single unified field,[40] Beyond this "grand unification", it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 1019 GeV. However -and while special relativity is parsimoniously incorporated into quantum electrodynamics- the expanded general relativity, currently the best theory describing the gravitation force, has not been incorporated into quantum theory at all.

[edit] Relativity and quantum mechanics
Main articles: Quantum gravity and Theory of everything

Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated empirical evidence and while they do not directly contradict each other theoretically (at least with regard to primary claims), they are resistant to being incorporated within one cohesive model.[41]

Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly contributing to the field, he did not accept the more philosophical consequences and interpretations of quantum mechanics, such as the lack of deterministic causality and the assertion that a single subatomic particle can occupy numerous areas of space at one time. He also was the first to notice some of the apparently exotic consequences of entanglement and used them to formulate the Einstein-Podolsky-Rosen paradox, in the hope of showing that quantum mechanics had unacceptable implications. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that Einstein's assumption was correct, but had to be completed by hidden variables and thus based on wrong philosophical assumptions. According to the paper of J. Bell and the Copenhagen interpretation (the common interpretation of quantum mechanics by physicists for decades), and contrary to Einstein's ideas, quantum mechanics was

neither a "realistic" theory (since quantum measurements do not state pre-existing properties, but rather they prepare properties)
nor a local theory (essentially not, because the state vector determines simultaneously the probability amplitudes at all sites, ).
The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner, although the two particles can be an arbitrary distance apart; however, this effect does not violate causality, since no transfer of information happens. These experiments are the basis of some of the most topical applications of the theory, quantum cryptography, which works well, although at small distances of typically 1000 km, being on the market since 2004.

Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology and physicists search for an elegant "Theory of Everything". Thus, resolving the inconsistencies between both theories has been a major goal of twentieth- and twenty-first-century physics. Many prominent physicists, including Professor Stephen Hawking, have labored in the attempt to discover a theory underlying everything, combining not only different models of subatomic physics, but also deriving the universe's four forces —the strong force, electromagnetism, weak force, and gravity— from a single force or phenomenon. One of the leading minds in this field is Edward Witten, a theoretical physicist who formulated the groundbreaking M-theory, which is an attempt at describing the supersymmetrical based string theory.

[edit] Applications
Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behaviour of the subatomic particles that make up all forms of matter—electrons, protons, neutrons, photons and others—can often only be satisfactorily described using quantum mechanics. Quantum mechanics has strongly influenced string theory, a candidate for a theory of everything (see reductionism) and the multiverse hypothesis. It is also related to statistical mechanics.

Quantum mechanics is important for understanding how individual atoms combine covalently to form chemicals or molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. (Relativistic) quantum mechanics can in principle mathematically describe most of chemistry. Quantum mechanics can provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others, and by approximately how much.[42] Most of the calculations performed in computational chemistry rely on quantum mechanics.[43]

Much of modern technology operates at a scale where quantum effects are significant. Examples include the laser, the transistor, the electron microscope, and magnetic resonance imaging. The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics.

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances.

In many devices, even the simple light switch, quantum tunneling is vital, as otherwise the electrons in the electric current could not penetrate the potential barrier made up, in the case of the light switch, of a layer of oxide. Flash memory chips found in USB drives also use quantum tunneling to erase their memory cells.

[edit] Philosophical consequences
Main article: Interpretation of quantum mechanics
Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpretations. Even fundamental issues such as Max Born's basic rules concerning probability amplitudes and probability distributions took decades to be appreciated.

The Copenhagen interpretation, due largely to the Danish theoretical physicist Niels Bohr, is the interpretation of quantum mechanics most widely accepted amongst physicists. According to it, the probabilistic nature of quantum mechanics predictions cannot be explained in terms of some other deterministic theory, and does not simply reflect our limited knowledge. Quantum mechanics provides probabilistic results because the physical universe is itself probabilistic rather than deterministic.

Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement (this dislike is the source of his famous quote, "God does not play dice with the universe."). Einstein held that there should be a local hidden variable theory underlying quantum mechanics and that, consequently, the present theory was incomplete. He produced a series of objections to the theory, the most famous of which has become known as the EPR paradox. John Bell showed that the EPR paradox led to experimentally testable differences between quantum mechanics and local realistic theories. Experiments have been performed confirming the accuracy of quantum mechanics, thus demonstrating that the physical world cannot be described by local realistic theories.[44] The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view.

The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a "multiverse" composed of mostly independent parallel universes.[45] This is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the collapse of the wave packet: All the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical (not just formally mathematical, as in other interpretations) quantum superposition. (Such a superposition of consistent state combinations of different systems is called an entangled state.) While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can observe only the universe, i.e. the consistent state contribution to the mentioned superposition, we inhabit. Everett's interpretation is perfectly consistent with John Bell's experiments and makes them intuitively understandable. However, according to the theory of quantum decoherence, the parallel universes will never be accessible to us. This inaccessibility can be understood as follows: once a measurement is done, the measured system becomes entangled with both the physicist who measured it and a huge number of other particles, some of which are photons flying away towards the other end of the universe; in order to prove that the wave function did not collapse one would have to bring all these particles back and measure them again, together with the system that was measured originally. This is completely impractical, but even if one could theoretically do this, it would destroy any evidence that the original measurement took place (including the physicist's memory).

[edit] See also
Bohm interpretation
Copenhagen interpretation
Correspondence rules
EPR paradox
Fine-structure constant
Interpretation of quantum mechanics
Introduction to quantum mechanics
Many-worlds interpretation
Measurement in quantum mechanics
Measurement problem
Photon dynamics in the double-slit experiment
Photon polarization
Quantum chaos
Quantum chemistry
Quantum chemistry computer programs
Quantum chromodynamics
Quantum computers
Quantum decoherence
Quantum electrochemistry
Quantum electronics
Quantum field theory
Quantum information
Quantum mind[46]
Quantum optics
Quantum pseudo-telepathy
Quantum thermodynamics
Quantum triviality
Quantum Zeno effect
Quantum spectrum
Quasi-set theory
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
Schrödinger's cat
Theoretical and experimental justification for the Schrödinger equation
Theoretical chemistry
Trojan wave packet"

 

 

Quantum mind - Wikipedia, the free encyclopedia Quantum mind theories are based on the premise that quantum mechanics is necessary to fully understand the mind and brain, particularly concerning an ...

Introduction - Motivation - Examples of theories - Ongoing Debate
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Quantum mind
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Quantum mind theories are based on the premise that quantum mechanics is necessary to fully understand the mind and brain, particularly concerning an explanation of consciousness. This approach is considered a minority opinion in science, although it does have the support of the well-known Roger Penrose, who has proposed a quantum mind theory in conjunction with Stuart Hameroff. Karl H. Pribram and Henry Stapp have also proposed variations.

Contents [hide]
1 Introduction
2 Motivation
2.1 Consciousness Banished
2.2 Minimization of Mystery
3 Examples of theories
3.1 David Bohm
3.2 Gustav Bernroider
3.3 David Chalmers
3.4 Roger Penrose
3.5 Evan Harris Walker
3.6 Henry Stapp
3.7 Quantum Brain Dynamics
3.8 Quantum Evidence
4 Ongoing Debate
4.1 Science
4.2 Philosophy
5 See also
6 Notes
7 References
8 Further reading
9 External links

[edit] Introduction
The quantum mind hypothesis proposes that classical mechanics cannot fully explain consciousness and suggests that quantum mechanical phenomena such as quantum entanglement and superposition may play an important part in the brain's function and could form the basis of an explanation of consciousness.

Supporters of the quantum mind hypothesis have not submitted any evidence to support its claims for peer review, but the hypothesis has also not been falsified. As such, the hypothesis is still in its early phases.

[edit] Motivation

[edit] Consciousness Banished
A common argument underlying the quantum mind thesis is that classical mechanics cannot explain consciousness, if only because Galileo and Newton (together with later thinkers, viz.: Locke, Hobbes and Descartes) excluded the secondary qualities from the physical world[citation needed].

Fritjof Capra writes:

To make it possible for scientists to describe nature mathematically, Galileo postulated that they should restrict themselves to studying the essential properties of material bodies—shapes, numbers, and movement—which could be measured and quantified. Other properties, like color, sound, taste, or smell, were merely subjective mental projections which should be excluded from the domain of science. [1]

Proponents of the Quantum mind state that perceived qualities such as sound, taste and smell are an essential part of the human experience and therefore cannot be discounted. They posit that classical mechanics fails to account for the experience of such phenomena. Similarly, they hypothesize that the internal experiences of consciousness, such as dreaming and memory, all of which are 'part and parcel' of everyday human experience remain unaccounted for.

[edit] Minimization of Mystery
Philosopher David Chalmers states that the motivation for Quantum Mind theories is: "a Law of Minimization of Mystery: consciousness is mysterious and quantum mechanics is mysterious, so maybe the two mysteries have a common source. Nevertheless, quantum theories of consciousness suffer from the same difficulties as neural or computational theories." [2]

[edit] Examples of theories

[edit] David Bohm
David Bohm took the view that quantum theory and relativity contradicted one another, and that this contradiction implied that there existed a more fundamental level in the physical universe[3][dead link]. He claimed that both quantum theory and relativity pointed towards this deeper theory. This more fundamental level was supposed to represent an undivided wholeness and an implicate order, from which arose the explicate order of the universe as we experience it.

Bohm's implicate order applies both to matter and consciousness, and he proposed that it could explain the relationship between them. Mind and matter are here seen as projections into our explicate order from the underlying reality of the implicate order. Bohm claims that when we look at the matter in space, we can see nothing in these concepts that helps us to understand consciousness.

In Bohm's scheme there is a fundamental level where consciousness is not distinct from matter. Bohm's view of consciousness is connected to Karl Pribram's holographic conception of the brain [4][5][dead link]. Pribram regards sight and the other senses as lenses without which the other senses would appear as a hologram. Pribram proposes that information is recorded all over the brain, and that it is enfolded into a whole, similar to a hologram. It is suggested that memories are connected by association and manipulated by logical thought. If the brain is also receiving sensory input all these are proposed to unite in overall experience or consciousness.

In trying to describe the nature of consciousness, Bohm discusses the experience of listening to music. He thinks that the feeling of movement and change that make up our experience of music derives from both the immediate past and the present being held in the brain together, with the notes from the past seen as transformations rather than memories. The notes that were implicate in the immediate past are seen as becoming explicate in the present. Bohm compares this to consciousness emerging from the implicate order.

Bohm sees the movement, change or flow and also the coherence of experiences such as listening to music as a manifestation of the implicate order. He claims to derive evidence for this from the work of Piaget[6] in studying infants. He claims that these studies show that young children have to learn about time and space, because they are part of the explicate order, but have a 'hard-wired' understanding of movement because it is part of the implicate order. He compares this 'hard-wiring' to Chomsky's theory that grammar is 'hard-wired' into young human brains. In his writings, Bohm never proposed any specific brain mechanism by which his implicate order could emerge in a way that was relevant to consciousness.

[edit] Gustav Bernroider
Recent papers by physicist Gustav Bernroider, have indicated that he thinks that Bohm's implicate-explicate structure can account for the relationship between neural processes and consciousness[7]. In a paper published in 2005 Bernroider elaborated his proposals for the physical basis of this process[8]. The main thrust of his paper was the argument that quantum coherence may be sustained in ion channels for long enough to be relevant for neural processes and the channels could be entangled with surrounding lipids and proteins and with other channels in the same membrane. Ion channels regulate the electrical potential across the axon membrane and thus play a central role in the brain's information processing.

Bernroider bases his work on recent studies of the potassium (K+)ion channel in its closed state and draws particularly on the atomic-level spectroscopy work of the MacKinnon group [9][10][11][12][13]. The ion channels have a filter region which allows in K+ ions and bars other ions. These studies show that the filter region has a framework of five sets of four oxygen atoms, which are part of the carboxyl group of amino-acid molecules in the surrounding protein. These are referred to as binding pockets. Two K+ ions are trapped in the selection filter of the closed ion channel. Each of these ions is electrostatically bound to two sets of oxygen atoms or binding pockets, involving eight oxygen atoms in total. Both ions in the channel oscillate between two configurations.

Bernroider uses this recently revealed structure to speculate about the possibility of quantum coherence in the ion channels. Bernroider and co-author Sisir Roy's calculations suggested to them that the behaviour of the ions in the K channel could only be understood at the quantum level. Taking this as their starting point, they then ask whether the structure of the ion channel can be related to logic states. Further calculations lead them to suggest that the K+ ions and the oxygen atoms of the binding pockets are two quantum-entangled sub-systems, which they then equate to a quantum computational mapping. The ions that are destined to be expelled from the channel are proposed to encode information about the state of the oxygen atoms. It is further proposed the separate ion channels could be quantum entangled with one another.

[edit] David Chalmers
The philosopher David Chalmers has speculated on a number of ways in which quantum mechanics might relate to consciousness.

One possibility is that instead of postulating novel properties, physics might end up appealing to consciousness itself, in the way that some theorists but not all, hold that quantum mechanics does. [14]

The collapse dynamics leaves a door wide open for an interactionist interpretation. [15]

The most promising version of such an interpretation allows conscious states to be correlated with the total quantum state of a system, with the extra constraint that conscious states (unlike physical states) can never be superposed. In a conscious physical system such as a brain, the physical and phenomenal states of the system will be correlated in a (nonsuperposed) quantum state. Upon observation of a superposed external system, Schrödinger evolution at the moment of observation would cause the observed system to become correlated with the brain, yielding a resulting superposition of brain states and so (by psychophysical correlation) a superposition of conscious states. But such a superposition cannot occur, so one of the potential resulting conscious states is somehow selected (presumably by a nondeterministic dynamic principle at the phenomenal level). The result is that (by psychophysical correlation) a definite brain state and a definite state of the observed object are also selected. [16]

If physics is supposed to rule out interactionism, then careful attention to the detail of physical theory is required. [17]

[edit] Roger Penrose
In The Emperor's New Mind and Shadows of the Mind, Penrose argues that

1. Humans have abilities, particularly mathematical ones, that no algorithmic computer (specifically Turing machine) could have, because computers are limited by Gödel's incompleteness theorem. In other words, he believes humans are hypercomputers. (The argument was originally due to John Lucas.)

Gödel demonstrated that with any recursively enumerable set of axioms capable of expressing Peano arithmetic, it was possible to produce a statement that was obviously true, but could not be proved by the axioms. The theorem enjoys general acceptance in the mathematical community[18].

Penrose, however, built a further and highly controversial argument on this theorem. He argued that the theorem showed that the brain had the ability to go beyond what can be demonstrated by mathematical axioms, and therefore there is something within the functioning of the brain that is not based on an algorithm (a system of calculations). A computer is just a system of algorithms, and Penrose claimed that Gödel's theorem demonstrated that brains could perform functions that no computer could perform.

Penrose is not interested in explaining phenomenal consciousness, qualia, generally regarded as the most mysterious feature of consciousness, but instead focuses mainly on the cognitive powers of mathematicians.

These assertions have been vigorously contested by many critics and notably by the philosophers Churchland and Grush[19][20]. The theory has been much criticised [21] [22] [23].

2. This would require some new physics. Penrose postulates that the currently unknown process underlying quantum collapse supplies the non-algorithmic element.

The random choice of, for instance, the position of a particle, which is involved in the collapse of the wave function was the only physical process that Penrose could find, which was not based on an algorithm. However, randomness was not a promising basis for the quality of mathematical judgement highlighted by his Gödel theorem argument.

But Penrose went on to propose that when the wave function did not collapse as a result of a measurement or decoherence in the environment, there could be an alternative form of wave function collapse, which he called objective reduction (OR). In this, each quantum superposition has its own space time geometry. When these become separated by more than the Planck length, they are affected by gravity, become unstable and collapse. OR is strikingly different both from the traditional orthodoxy of Niels Bohr's Copenhagen interpretation of quantum theory and from some more modern theories which avoid wave function collapse altogether such as Many-worlds interpretation or some forms of Quantum decoherence theory.

Penrose further proposes that OR is neither random nor governed by an algorithm, but is 'non-computational', selecting information embedded in the fundamental level of space time geometry.

3. Collapse requires a coherent superposed state to work on. Penrose borrows Stuart Hameroff's proposal about microtubules to supply this.

Initially, Penrose had lacked any detailed proposals for how OR could occur in the brain. Later on cooperation with Stuart Hameroff [24] supplied this side of the theory. Microtubules were central to Hameroff's proposals. These are the core element of the cytoskeleton, which provides a supportive structure and performs various functions in body cells. In additions to these functions, it was now proposed that the microtubules could support macroscopic quantum features known as Bose-Einstein condensates. It was also suggested that these condensates could link with other neurons via gap junctions. This is claimed to permit quantum coherence to extend over a large area of the brain. It is suggested that when one of these areas of quantum coherence collapses, there is an instance of consciousness, and the brain has access to a non-computational process embedded in the fundamental level of space time geometry.

At the same time, it was postulated that conventional synaptic activity influences and is influenced by the activity in the microtubules. This part of the process is referred to as 'orchestration' hence the theory is called Orchestrated Objective Reduction or more commonly Orch OR.

Hameroff's proposals like those of Penrose attracted much criticism. However the most cogent attack on Orch OR and quantum mind theories in general was the view that conditions in the brain would lead to any quantum coherence decohering too quickly for it to be relevant to neural processes. This general criticism is discussed in the Science section below.

[edit] Evan Harris Walker
Information theory is concerned with the capacity to contain or carry information. Is there such a thing as a conscious field and a conscious channel capacity?

Information theory is concerned with the measurement of information in terms of logarithmic probability—how many bits of information does it take to represent a certain type of information such as, let’s say, the letter “T” in print. Since we don’t know all the possible permutations or “combinations” of such a question we use statistical probability in order to be very accurate in our measurements. We add up all the logarithmic contributions of each possible symbol being measured in terms of its chance of occurrence. It is expressed as log₂P. This gives us an informational field potential.

A physicist, Evan Harris Walker developed a scientific theory about how the brain might, at quantum levels, process information. In his book, The Physics of Consciousness, he adds log₂P to Schrödinger’s equation. What he demonstrates mathematically is that when information is measured by consciousness and will channel capacities in terms of a closed loop, it forces one real solution only when one probable state happens and all other possible states disappear. He offers/proposes physical evidence that this process is occurring in the brain.

[edit] Henry Stapp
Physically, Henry Stapp's approach is aligned with objective collapse theory, in that the deterministic evolution of the wave function, and its indeterministic collapse are seen as two real and ontologically distinct phenomena. Collapse events occurring within the brain — the mind's observation or measurement of the brain — are particularly important. Since Stapp sees collapse as a mental process and the deterministic evolution of brain states as physical, his approach is philosophically aligned with interactionist dualism. The process by which collapse selects an actuality from a set of possibilities is seen by Stapp as literally a process of choice, and not merely a random dice-throw. His approach has implications with regard to time. Since the future depends on decisions in the present, it is not pre-existing, as in the block universe theory; rather there is an evolving universe in which subjects participate, as in Whitehead's metaphysics. [25]

Stapp envisages consciousness as exercising top-level control over neural excitation in the brain. Quantum brain events are suggested to occur at the whole brain level, and are seen as being selected from the large-scale excitation of the brain. The neural excitations are viewed as a code, and each conscious experience as a selection from this code. The brain, in this theory, is proposed to be a self-programming computer with a self-sustaining input from memory, which is itself a code derived from previous experience. This process results in a number of probabilities from which consciousness has to select. The conscious act is a selection of a piece of top-level code, which then exercises ongoing control over the flow of neural excitation. This process refers to the top levels of brain activity involved with information gathering, planning and the monitoring of the execution of plans. Conscious events are proposed to be capable of grasping a whole pattern of activity, thus accounting for the unity of consciousness, and providing a solution to the 'binding problem'.

Stapp's version of the conscious brain is proposed to be a system that is internally determined in a way that cannot be represented outside the system, whereas for the rest of the physical universe an external representation plus a knowledge of the laws of physics allows an accurate prediction of future events.

Stapp proposes that the proof of his theory requires the identification of the neurons that provide the top-level code and also the process by which memory is turned into additional top-level code.

[edit] Quantum Brain Dynamics
The ideas behind QBD derived originally from the physicists, Hiroomi Umezawa[26], and Herbert Frohlich[27] in the 1960s. In recent decades these ideas have been elaborated and given greater prominence by younger physicists such as Mari Jibu[28], Kunio Yasue[29] and Giuseppe Vitiello[30]. In quantum brain dynamics (QBD), the electrical dipoles of the water molecules that constitute 70% of the brain are proposed to constitute a quantum field, known here as the cortical field. The quanta of this field are described as corticons. In the theory, this field interacts with quantum coherent waves generated by biomolecules in the neurons and propagating along the neuronal network.

Frohlich is the source of the idea that quantum coherent waves could be generated in the neuronal network. Frohlich argued that it was not clear how order could be sustained in living systems given the disruptive influence of the fluctuations in biochemical processes. He viewed the electric potential across the neuron membrane as the observable feature of some form of underlying quantum order. His studies claimed to show that with an oscillating charge in a thermal bath, large numbers of quanta may condense into a single state known as a Bose condensate. This state allows long-range correlation amongst the dipoles involved. Further to this, biomolecules were proposed to line up along actin filaments (part of the cytoskeleton) and dipole oscillations propagate along the filaments as quantum coherent waves. This now has some experimental support in the form of confirmation that biomolecules with high electric dipole moment have been shown to have a periodic oscillation[31]. Vitiello also argues that the ordered chains of chemical reactions on which biological tissues depend would collapse without some form of quantum ordering, which in QBD is described by quantum field theory rather than quantum mechanics.

Vitiello provides citations, which are claimed to support his view of biological tissue. These include studies of radiation effect on cell growth[32],response to external stimuli[33], non-linear tunnelling[34],coherent nuclear motion in membrane proteins[35],optical coherence in biological systems[36], energy transfer via solitons and coherent excitations[37].

QBD proposes that the cortical field not only interacts with, but also to a good extent controls the neuronal network. It suggests that biomolecular waves propagate along the actin filaments in the area of the cell membranes and dendritic spines. The waves derive energy from ATP molecules stored in the cell membrane and control the ion channels, which in turn regulate the flow of signals to the synapses. Vitiello claims that QBD does not require quantum oscillations to last as long as the actual time to decoherence.

The proponents of QBD differ somewhat as the exact way in which it produces consciousness. Jibu and Yasue think that the interaction between the energy quanta of the cortical field and the biomolecular waves of the neuronal network, particularly the dendritic part of the network, is what produces consciousness. On the other hand, Vitiello thinks that the quantum states involved in QBD produce two poles, a subjective representation of the external world and a self. This self opens itself to the representation of the external world. Consciousness is, in this theory, not in either the self or the external representation, but between the two in the opening of one to the other.

[edit] Quantum Evidence
Concerning the question where quantum events could be effective in the complex cortical structure and brain dynamics, it is outlined that professors Friedrich Beck and John C. Eccles developed a model for quantal emission process at the synaptic cleft with reasonable results. These authors also discussed in detail the problem of elementary microscopic processes in protein complexes able to survive thermal fluctuations. Quantum evidence was also reached by the physicist Evan Harris Walker. Under the mental-psychological profile, in years from 2003 to 2009, the professors Elio Conte, Andrei Yuri Khrennikov, Orlando Todarello, Antonio Federici, Joseph P. Zbilut, have performed a number of experiments reaching evidence on possible existence of quantum interference effects on mental states during human perception and cognition of ambiguous figures. See further reading[13, 14, 15, 16]. These authors have also realized theoretical contributions on the analysis of quantum interference effects in mental states, and on time dynamics of cognitive entities[13, 14, 15, 16].

Physicists at the University of California, Berkeley believe they have discovered that green plants perform quantum computation in order to capture the sun's light through photosynthesis—evidence of quantum coherence in a living system.[38] Stuart Hameroff noted, in October 2000, that quantum coherence—although, by its mere occurrence in the brain not sufficient to prove its supposed central role in consciousness—had nevertheless been observed. This, he claimed, was significant because so much of the criticism of his model had "come under sharp criticism due to the issue of decoherence, and the question of whether quantum processes of significance can exist in the brain at physiological temperature." (Quantum Mind archives, October 2000 - (11.))

[edit] Ongoing Debate

[edit] Science
The main argument against the quantum mind proposition is that the structures of the brain are much too large for quantum effects to be important. It is impossible for coherent quantum states to form for very long in the brain and impossible for them to exist at scales on the order of the size of neurons. Price, for example, says that quantum effects rarely or never affect human decisions and that classical physics determines the behaviour of Neurons.

In quantum terms each neuron is an essentially classical object. Consequently quantum noise in the brain is at such a low level that it probably doesn't often alter, except very rarely, the critical mechanistic behaviour of sufficient neurons to cause a decision to be different than we might otherwise expect. (...)

—Michael Clive Price[1]
This does not imply that classical mechanics can explain consciousness, but that quantum effects including superposition and entanglement are insignificant.

One well-known critic of the quantum mind is Max Tegmark. Based on his calculations, Tegmark concluded that quantum systems in the brain decohere quickly and cannot control brain function, "This conclusion disagrees with suggestions by Penrose and others that the brain acts as a quantum computer, and that quantum coherence is related to consciousness in a fundamental way"[39]

Proponents of quantum consciousness theories have sought to defend them against Tegmark's criticism. In respect of QBD, Vitiello has argued that Tegmark's work applies to theories based on quantum mechanics but not to those such as QBD that are based on quantum field theory. In respect of Penrose and Hameroff's Orch OR theory, Hameroff along with Hagan and Tuszynski replied to Tegmark[40]. They claimed that Tegmark based his calculations on a model that was different from Orch OR. It is argued that in the Orch OR model the microtubules are shielded from decoherence by ordered water. Energy pumping as a result of thermal disequilibrium, Debye layer screening and quantum error correction, deriving from the geometry of the microtubule lattice are also proposed as possible sources of shielding. Similarly, in his extension of Bohm's ideas, Bernroider has claimed that the binding pockets in the ion selection filters could protect against decoherence[41]. So far, however, there has been no experimental confirmation of the ability of the features mentioned above to protect against decoherence.

[edit] Philosophy
Another line of criticism is that no physical theory is well suited to explaining consciousness, particularly in its most problematical form, phenomenal consciousness or qualia, known as the hard problem of consciousness.[42] It is not so much that colours and tastes and feelings—qualia or secondary qualities—have been deliberately banished, but more that they cannot be captured in any mathematical description, which means they cannot be explicitly represented in physics, since all physical theory is expressed in mathematical language (as explained in Eugene Wigner's famous paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences). If no physical theory can express qualia, no physical theory can fully explain consciousness. Replacing the mathematical apparatus of classical physics with the mathematical apparatus of quantum mechanics is therefore of no help in understanding consciousness, and indeed there is no known example of a quantum equation which encapsulates a taste or colour.

As David Chalmers puts it:

Nevertheless, quantum theories of consciousness suffer from the same difficulties as neural or computational theories. Quantum phenomena have some remarkable functional properties, such as nondeterminism and nonlocality. It is natural to speculate that these properties may play some role in the explanation of cognitive functions, such as random choice and the integration of information, and this hypothesis cannot be ruled out a priori. But when it comes to the explanation of experience, quantum processes are in the same boat as any other. The question of why these processes should give rise to experience is entirely unanswered. [2]

Other philosophers, such as Patricia and Paul Churchland and Daniel Dennett[43] reject the idea that there is anything puzzling about consciousness in the first place.

[edit] See also
Bohm interpretation of quantum mechanics
Consciousness causes collapse interpretation of quantum mechanics
Electromagnetic theories of consciousness
Evolutionary neuroscience
Hard problem of consciousness
Holonomic brain theory
Mechanism (philosophy)
Roger Penrose's Quantum Mind theory
Quantum brain dynamics
Theory of mind
[edit] Notes
1.^ Capra, F. The Turning Point
2.^ a b Chalmers, D. Facing up to the Problem of Consciousness
3.^ Bohm 2005
4.^ Pribram 1991
5.^ #Reference-idPribram1999
6.^ Piaget 1956
7.^ Bernroider 2003
8.^ Bernroider 2005
9.^ Jiang 2003
10.^ Jiang 2003
11.^ Zhou 2001
12.^ Morais-Cabral 2001
13.^ Doyle 1998
14.^ Chalmers, D. Consciousness and its Place in Nature
15.^ Chalmers, D. Consciousness and its Place in Nature
16.^ Chalmers, D. Consciousness and its Place in Nature
17.^ Chalmers, D. Consciousness and its Place in Nature
18.^ Nagel 1958
19.^ #Reference-idGrush1995
20.^ Churchland 1996
21.^ Georgiev, Danko. Falsifications of Hameroff-Penrose Orch OR Model of Consciousness and Novel Avenues for Development of Quantum Mind Theory
22.^ Review of Penrose, The Emperor's New Mind
23.^ Puttnam, H. Review of Penrose, The Emperor's New Mind
24.^ Hameroff 1987
25.^ stapp, H. Quantum Approaches to Consciousness
26.^ Ricciardi 1967
27.^ Frohlich 1968
28.^ Jibu 1995
29.^ Jibu 1995
30.^ Vitiello 2001
31.^ Gray 1989
32.^ Grundler 1992
33.^ Kaiser 1988
34.^ Huth 1984
35.^ Vos 1993
36.^ Li 1983
37.^ Huth 1989
38.^ Sources:
David Biello (2007-04-13). "When It Comes to Photosynthesis, Plants Perform Quantum Computation". Scientific American. http://www.sciam.com/article.cfm?id=when-it-comes-to-photosynthesis-plants-perform-quantum-computation.
"Quantum Secrets of Photosynthesis Revealed". Research News Berkeley lab. 2007-04-12. http://www.lbl.gov/Science-Articles/Archive/PBD-quantum-secrets.html.
"New Quantum Secrets of Photosynthesis". Berkeley lab. 2007-08-06. http://www.lbl.gov/Science-Articles/Archive/sabl/2007/Jul/quantumSecrets.html.
39.^ Importance of quantum decoherence in brain processes by Max Tegmark in Phys. Rev. E (2000) Volume 61, pages 4194 - 4206.
40.^ Hagan 2002
41.^ Bernroider 2005
42.^ Saul-Paul Sirag"Consciousness:A Hyperspace View"
43.^ Dennet, D. Facing Backwards on the Problem of Consciousness

[edit] References
Bohm, D.(1980) - Wholeness and the Implicate Order - John Benjamins [2]
Pribram, K.(1991) - Brain and Perception -Lawrence Erlbaum - [3]
Pribram, K.(1999) - Brain and the composition of conscious experience - Journal of Consciousness Studies,6,(5), pp.12-18 [4]
Piaget, J.(1956) - The Origin of Intelligence in the Child - Routledge & Kegan Paul [5]
Bernroider, G.(2003) - Quantum neurodynamics and the relation to conscious experience - Neuroquantology, 2, pp.163-8 [6]
Bernroider, G.& Roy, S.(2005) - Quantum entanglement of K ions, multiple channel states and the role of noise in the brain - SPIE Vol. 5841-29, pp.205-14 [7]
Jiang, Y.,MacKinnon,R.et al. (2003) - X-ray structure of a voltage dependent K+ channel - Nature,423,pp.33-41 [8]
Jiang, Y.MacKinnon,R. et al. (2003) - The principle of gating charge movement in a voltage dependent K+ channel - Nature 423,pp.42-8 [9]
Zhou, Y.,Morais-Cabral, A.,Kaufman, A.& MacKinnon,R.(2001) - Chemistry of ion coordination and hydration revealed in in K+ channel-Fab complex at 2.0 A resolution - Nature,414,pp.43-8 [10]
Morais-Cabral, H.,Zhou, H.& MacKinnon,R.(2001) - Energetic optimisation of ion conduction rates by the K+ selectivity filter - Nature,414,pp.37-42 [11]
Doyle, D.,MacKinnon,R.et al. (1998) - The structure of the potassium channel: Molecular basis of K+ conduction and selectivity - Science,280,pp.69-76 [12]
Ricciardi, L.& Umezawa, H.(1967) - Kybernetik,4,pp.44-48 [13]
Frohlich, H. (1968) - Long range coherence and energy storage in biological systems - International Journal of Quantum Chemistry,2,pp.641-649 [14]
Jibu, M.& Yasue, K (1995) - Quantum Brain Dynamics and Consciousness:Advances in Consciousness Research - John Benjamins
Vitiello, G. (2001) - My Double Unveiled; Advances in Consciousness - John Benjamins [15]
Gray, C.& Singer, W.(1989) - Stimulus specific neuronal oscillations in orientation of cat visual cortex - Proceedings of the National Academy of Science,86,pp.1698-1702 [16]
Grundler & Kaiser (1992) - Experimental evidence for coherent excitations correlated with cell growth - Nanobiology,1,pp.163-176
Kaiser, F. (1988) - Theory of non-linear excitations - In: Frohlich, H. Ed., Biological coherence and response to external stimuli, p.25-48 - Springer Verlag
Huth et al. (1984) - Nonlinear tunneling barrier at high frequencies and their possible logic processing function in biological membranes - In: Adey, R.& Lawrence, A. Eds., Non-linear dynamics in biological systems, pp.227-241 - Plenum
Vos M. et al. (1993) - Visualisation of coherent nuclear motion in a membrane protein by femtosecond spectroscopy - Nature,363,pp.320-335 [17]
Li, K.et al. (1983) - Indications of optical coherence in biological systems - In: Frohlich, H.& Kremer, F. Eds, Coherent excitations in biological systems, pp.117-22 - Springer Verlag
Huth, G.,Gutmann, F.& Vitiello, G.(1989) - The lifetime of coherent excitations - Phys. Lett, A 154,pp.339-42 [18]
Hagan, S.,Hameroff, S.& Tuszynski, J.(2002) - Quantum computation in brain microtubules? Decoherence and biological feasibility - Physical Reviews, E65: 061901 [19]
Nagel, E.& Newman, J. (1958) - Godel's Proof - Routledge
Grush, R.& Churchland, P. (1995) - Gap's in Penrose's Toilings - Journal of Consciousness Studies, 2,(1), pp.10-29 [20]
Churchland, P. (1996) - The Hornswoggle Problem - Journal of Consciousness Studies,3,Nos5-6,pp.218-20 [21]
Hameroff, S. (1987) - Ultimate Computing - Elsevier [22]

[edit] Further reading
Bennett, Charles H., Shor, Peter W., Smolin, John A. and Thapliyal, Ashish V. Entanglement-Assisted Classical Capacity of Noisy Quantum Channels, Phys. Rev. Lett. 83, 3081–3084 (1999). [23]
Conte, Elio, Todarello, Orlando, Federici, Antonio, Vitiello, Francesco, Lopane, Michele, Khrennikov, Andrei and Joseph P. Zbilut: Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics. Chaos, Solitons and Fractals 31 (2007) 1076-1088 [24]
Conte Elio, Khrennikov Yuri Andrei, Todarello Orlando,Federici Antonio, Zbilut Joseph P.: Mental States Follow Quantum Mechanics during Perception and Cognition of Ambiguous Figures. Open Systems & Information Dynamics, (2009),Vol.16, No.1,1-17; available on line PhilPapers,[25].
Conte Elio, Khrennikov Yuri, Todarello Orlando, Federici Antonio, Zbilut Joseph P: On the Existence of Quantum Wave Function and Quantum Interference Effects in Mental States: An Experimental Confirmation during Perception and Cognition in Humans. NeuroQuantology, (2009), First issue 2009 - available on line.[26]
Conte Elio: Exploration of Biological Function by Quantum Mechanics. Proceedings 10th International Congress on Cybernetics, 1983;16-23, Namur-Belgique.
Conte Elio: Testing Quantum Consciousness. NeuroQuantology (2008); 6 (2): 126-139.[27]
Conte Elio, Khrennikov Yuri Andrei, Todarello Orlando, Federici Antonio, Zbilut Joseph P.: A Preliminary Experimental Verification On the Possibility of Bell Inequality Violation in Mental States. NeuroQuantology, (2008); 6 (3): 214-221.[28]
Khrennikov Yuri Andrei: Quantum-like brain: Interference of minds. BioSystems 84, 225-241 (2006). [29]
Khrennikov Yuri Andrei: Information Dynamics in Cognitive, Psychological and Anomalous Phenomena. ser. Fundamental Theories of Physics, Kluwer Academic, 2004.
Conte Elio, Todarello Orlando, Federici Antonio, Zbilut Joseph P.: Mind States follow Quantum Mechanics during Perception and Cognition of Ambiguous Figures: a Final Experimental Confirmation. arXiv:0802.1835 [30]
Flanagan, Brian. Are Perceptual Fields Quantum Fields? [31]
Hodgson, David. The Mind Matters. Oxford University Press, 1993.
Koch, C. and Hepp, K., Quantum mechanics in the brain, Nature 440, 611 (30 March 2006)
Litt et al., Is the Brain a Quantum Computer?, Cognitive Science (2006) [32]
Lockwood, Michael. Mind, Brain and the Quantum. Cambridge, MA: Basil Blackwell Ltd., 1989.
McFadden, Johnjoe (2000) Quantum Evolution HarperCollins. ISBN 0-00-255948-X; ISBN 0-00-655128-9 . Final chapter on the quantum mind.
Pharoah, M.C. (online). Looking to quantum mechanics for a reductive explanation of the noumenon of consciousness Retrieved June.21 2008.
Schrödinger, Erwin. Mind and Matter. Cambridge University Press, 1959.
Weiss V, Weiss H (2003). The golden mean as clock cycle of brain waves. Chaos, Solitons and Fractals 18: 643-652. Full text
Weyl, Hermann. Mind and Nature, University of Pennsylvania Press, 1934.
Wigner, Eugene. "Physics and the Explanation of Life," in Foundations of Physics, vol. 1, 1970, pp. 34–45.

[edit] External links
Macroscopic quantum effects in biophysics and consciousness
NeuroQuantology Journal
The Science and Philosophy of Consciousness
Quantum Interconnectedness
Problem with Quantum Mind Theory
Center for Consciousness Studies, directed by Stuart Hameroff
Stuart Hameroff's Quantum Consciousness site
Online papers on the quantum mechanisms of consciousness
Spin & Consciousness Research
Stanford Univ. Encyclopedia of Philosophy critical survey article Harald Altmanspacher (2006) Quantum Approaches to Consciousness
Henry Stapp's collection of articles and papers
Quantumbionet
Quantum-Mind
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-
-
-
-
-
6
-
1
+
=
7
-
=
7
=
7
-
-
-
15
-
19
+
=
34
3+4
=
7
=
7
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
1
2
-
4
-
+
=
7
-
=
7
=
7
-
1
20
-
13
-
+
=
34
3+4
=
7
=
7
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
1
20
15
13
19
+
=
68
6+8
=
14
1+4
5
-
1
2
6
4
1
+
=
14
1+4
=
5
=
5
5
A
T
O
M
S
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
5
A
T
O
M
S
-
-
13
-
-
5
-
14
-
-
-
-
-
-
-
-
1+3
-
-
-
-
1+4
5
A
T
O
M
S
-
-
4
-
-
5
-
5
-
1
2
6
4
1
-
-
-
-
-
-
-
-
5
A
T
O
M
S
-
-
4
-
-
5
-
5

 

 

-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
-
6
-
+
=
6
-
=
6
=
6
-
-
-
-
15
-
+
=
15
1+5
=
6
=
6
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
2
-
4
+
=
7
-
=
7
=
7
-
-
1
20
-
13
+
=
34
3+4
=
7
=
7
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
20
15
13
+
=
49
4+9
=
13
1+3
4
-
-
1
2
6
4
+
=
13
1+3
=
4
=
4
-
4
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
3
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
32
4
A
T
O
M
-
-
13
-
-
4
-
13
3+2
-
-
-
-
-
-
-
1+3
-
-
-
-
1+3
5
4
A
T
O
M
-
-
4
-
-
4
-
4
-
-
1
2
6
4
-
-
-
-
-
-
-
-
5
4
A
T
O
M
-
-
4
-
-
4
-
4

 

 

-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
-
6
-
+
=
6
-
=
6
=
6
-
-
-
-
15
-
+
=
15
1+5
=
6
=
6
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
2
-
4
+
=
7
-
=
7
=
7
-
-
1
20
-
13
+
=
34
3+4
=
7
=
7
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
20
15
13
+
=
49
4+9
=
13
1+3
4
-
-
1
2
6
4
+
=
13
1+3
=
4
=
4
-
4
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
3
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
32
4
A
T
O
M
-
-
13
-
-
4
-
13
3+2
-
-
-
-
-
-
-
1+3
-
-
-
-
1+3
5
4
A
T
O
M
-
-
4
-
-
4
-
4
-
-
1
2
6
4
-
-
-
-
-
-
-
-
5
4
A
T
O
M
-
-
4
-
-
4
-
4

 

THE ATOM THE

 

THE ATUM THE

 

6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
4
occurs
x
1
=
4
6
A
T
U
M
-
-
10
-
-
4
-
10
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
6
A
T
U
M
-
-
1
-
-
4
-
2

 

 

THE HERMETICA

THE LOST WISDOM OF THE PHARAOHS

Timothy Freke & Peter Gandy

To the Memory of Giordano Bruno 1548 - 1600

Mundus Nihil Pulcherrimum

The World is a Beautiful Nothing

Page 23

"Although we have used the familiar term 'God' in the explanatory notes which accompany each chapter, we have avoided this term in the text itself. Instead we have used 'Atum - one of the ancient Egyptian names for the Supreme One God."

 

Page 45

The Being of Atum

"Atum is Primal Mind."

 

Page 45

The Being of Atum

Give me your whole awareness, and concentrate your thoughts, for Knowledge of Atum's Being requires deep insight, which comes only as a gift of grace.

It is like a plunging torrent of water whose swiftness outstrips any man who strives to follow it, leaving behind not only the hearer, but even the teacher himself.

To conceive of Atum is difficult.

To define him is impossible.

The imperfect and impermanent cannot easily apprehend the eternally perfected.

Atum is whole and conconstant.

In himself he is motionless, yet he is self-moving.

He is immaculate, incorruptible and ever-lasting.

He is the Supreme Absolute Reality. He is filled with ideas which are imperceptible to the senses, and with all-embracing Knowledge.

Atum is Primal Mind.

Page 46

He is too great to be called by the name 'Atum'. He is hidden, yet obvious everywhere.

His Being is known through thought alone, yet we see his form before our eyes.

He is bodiless, yet embodied in everything. There is nothing which he is not. He has no name, because all names are his name. He is the unity in all things, so we must know him by all names and call everything 'Atum'.

He is the root and source of all. Everything has a source, except this source itself, which springs from nothing.

Atum is complete like the number one, which remains itself whether multiplied or divided, and yet generates all numbers.

Atum is the Whole which contains everything. He is One, not two.

He is All, not many.

The All is not many separate things, but the Oneness that subsumes the parts.

The All and the One are identical.

You think that things are many when you view them as separate, but when you see they all hang on the One, /Page 47/ and flow from the One, you will realise they are united­linked together, and connected by a chain of Being from the highest to the lowest, all subject to the will of Atum.

The Cosmos is one as the sun is one, the moon is one and the Earth is one.

Do you think there are many Gods? That's absurd - God is one.

Atum alone is the Creator of all that is immortal, and all that is mutable.

If that seems incredible, just consider yourself. You see, speak, hear, touch, taste, walk, think and breathe.

It is not a different you who does these various things, but one being who does them all.

To understand how Atum makes all things, consider a farmer sowing seeds; here wheat - there barley,
now planting a vine - then an apple tree.

Just as the same man plants all these seeds, so Atum sows immortality in heaven and change on Earth.

Throughout the Cosmos he disseminates Life and movement­the two great elements that comprise Atum and his creation, and so everything that is.

Page 48

Atum is called 'Father' because he begets all things, and, from his example, the wise hold begetting children the most sacred pursuit of human life. Atum works with Nature, within the laws of Necessity, causing extinction and renewal, constantly creating creation to display his wisdom.

Yet, the things that the eye can see are mere phantoms and illusions.

Only those things invisible to the eye are real. Above all are the ideas of Beauty and Goodness.

Just as the eye cannot see the Being of Atum, so it cannot see these great ideas.

They are attributes of Atum alone, and are inseparable from him.

They are so perfectly without blemish that Atum himself is in love with them.

There is nothing which Atum lacks, so nothing that he desires.

There is nothing that Atum can lose, so nothing can cause him grief. Atum is everything.

Atum makes everything, and everything is a part of Atum.

Atum, therefore, makes himself.

This is Atum's glory - he is all-creative, and this creating is his very Being.

It is impossible for him ever to stop creating­for Atum can never cease to be.

Page 49

Atum is everywhere.

Mind cannot be enclosed, because everything exists within Mind.

Nothing is so quick and powerful.

Just look at your own experience. Imagine yourself in any foreign land, and quick as your intention you will be there!

Think of the ocean - and there you are.

You have not moved as things move, but you have travelled, nevertheless.

Fly up into the heavens - you won't need wings!

Nothing can obstruct you - not the burning heat of the sun, or the swirling planets.

Pass on to the limits of creation. Do you want to break out beyond the boundaries of the Cosmos?

For your mind, even that is possible.

Can you sense what power you possess? If you can do all this, then what about your Creator?

Try and understand that Atum is Mind.

This is how he contains the Cosmos. All things are thoughts which the Creator thinks."

 

 

-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
35
6
A
T
U
M
-
-
10
-
-
4
-
10
3+5
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
8
6
A
T
U
M
-
-
1
-
-
4
-
2

 

 

6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
4
occurs
x
1
=
4
6
A
T
U
M
-
-
10
-
-
4
-
10
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
6
A
T
U
M
-
-
1
-
-
4
-
2

 

 

 

Q
=
8
7
QUANTUM
107
26
8
M
=
4
9
MECHANICS
75
39
3
-
-
12
16
First Total
182
65
11
-
-
1+2
1+6
Add to Reduce
1+8+2
6+5
1+1
-
-
3
7
Second Total
11
11
2
-
-
-
-
Reduce to Deduce
1+1
1+1
-
-
-
3
7
Essence of Number
2
2
2

 

 

-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
8
-
5
9
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
-
14
-
-
-
-
-
-
-
8
-
14
9
-
19
+
=
64
6+4
=
10
1+0
1
=
1
-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
4
5
3
-
1
-
-
3
-
+
=
37
3+7
=
10
1+0
1
=
1
-
-
17
21
1
-
20
21
13
-
13
5
3
-
1
-
-
3
-
+
=
118
1+1+8
=
10
1+0
1
=
1
-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
13
5
3
8
1
14
9
3
19
+
=
182
1+8+2
=
11
1+1
2
=
2
-
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
+
=
65
6+5
=
11
1+1
2
=
2
-
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
``-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
``-
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
13
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
32
-
-
16
-
65
-
38
1+3
1+6
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3+2
-
-
1+6
-
6+5
-
3+8
4
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
11
-
11
-
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
-
-
-
-
-
-
-
1+1
-
1+1
4
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
2
-
2

 

 

16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
8
-
5
9
-
1
+
=
28
2+8
=
10
1+0
1
=
1
-
-
-
-
14
-
-
-
-
-
-
-
8
-
14
9
-
19
+
=
64
6+4
=
10
1+0
1
=
1
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
8
3
1
-
2
3
4
-
4
5
3
-
1
-
-
3
-
+
=
37
3+7
=
10
1+0
1
=
1
-
17
21
1
-
20
21
13
-
13
5
3
-
1
-
-
3
-
+
=
118
1+1+8
=
10
1+0
1
=
1
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
--
-
-
-
-
-
17
21
1
14
20
21
13
-
13
5
3
8
1
14
9
3
19
+
=
182
1+8+2
=
11
1+1
2
=
2
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
+
=
65
6+5
=
11
1+1
2
=
2
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
16
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
32
-
-
16
-
65
-
38
1+6
-
3
-
-
-
3
-
-
-
-
3
-
-
-
-
3
-
-
-
3+2
-
-
1+6
-
6+5
-
3+8
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
11
-
11
-
8
3
1
5
2
3
4
-
4
5
3
8
1
5
9
3
1
-
-
-
-
-
-
-
1+1
-
1+1
7
Q
U
A
N
T
U
M
-
M
E
C
H
A
N
I
C
S
-
-
5
-
-
7
-
2
-
2

 

 

Q
=
8
7
QUANTUM
107
26
8
E
=
5
6
ENERGY
74
38
2
-
-
13
13
First Total
181
64
10
-
-
1+3
1+3
Add to Reduce
1+8+1
6+4
1+0
-
-
4
4
Second Total
10
10
1
-
-
-
-
Reduce to Deduce
1+0
1+0
-
-
-
4
4
Essence of Number
1
1
1

 

 

Q
=
8
7
QUANTUM
107
26
8
E
=
5
8
ENERGIES
82
46
1
-
-
13
15
First Total
189
72
9
-
-
1+3
1+5
Add to Reduce
1+8+9
7+2
-
-
-
4
6
Second Total
18
9
9
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
6
Essence of Number
9
9
9

 

 

-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
9
-
1
+
=
20
2+0
=
2
-
2
-
2
-
-
-
-
-
14
-
-
-
-
-
14
-
-
-
9
-
19
+
=
56
5+6
=
11
1+1
2
-
2
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
5
-
5
9
7
-
5
-
+
=
52
5+2
=
7
-
7
=
7
-
-
17
21
1
-
20
21
13
-
5
-
5
18
7
-
5
-
+
=
133
1+3+3
=
7
-
7
-
7
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
5
14
5
18
7
9
5
19
+
=
189
1+8+9
=
18
1+8
9
-
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
+
=
72
7+2
=
9
-
9
=
9
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
2
-
``-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
``-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
-
5
-
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
``-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
-
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
6
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
39
-
-
15
-
72
-
45
=
1+5
-
-
-
-
-
-
-``
-
-
-
-
9
-
9
-
-
-
-
3+9
-
-
1+5
-
7+2
-
4+5
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
12
-
-
6
-
9
-
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
-
-
1+2
-
-
-
-
-
-
-
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
3
-
-
6
-
9
-
9

 

 

5
MESON
-
-
-
-
M+E
18
9
9
-
S+O+N
48
12
3
5
MESON
66
21
12
-
-
6+6
2+1
1+2
5
MESON
12
3
3
-
-
1+2
-
-
5
MESON
3
3
3

 

 

6
MESONS
-
-
-
-
M+E
18
9
9
-
S+O+N+S
67
13
4
6
MESONS
85
22
13
-
-
8+5
2+2
1+3
6
MESONS
13
4
4
8
THIRTEEN
99
45
9

 

 

11
SCHRODINGERS
139
67
4
3
CAT
24
6
6
14
First Total
163
73
10
1+4
Add to Reduce
1+6+3
7+3
1+0
5
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
5
Essence of Number
1
1
1

 

 

W
=
5
4
WAVE
51
15
6
P
=
7
8
PARTICLE
84
39
3
-
-
12
12
Add to Reduce
135
54
9
-
-
1+2
1+2
Reduce to Deduce
1+3+5
5+4
-
-
-
3
3
Essence of Number
9
9
9

 

 

5
WAVES
70
16
7
9
PARTICLES
103
40
4

 

 

4
WAVE
51
15
6
9
PARTICLES
103
40
4
13
First Total
154
55
10
1+3
Add to Reduce
1+5+4
5+5
1+0
4
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
4
Essence of Number
1
1
1

 

 

-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
-
9
-
9
-
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
-
9
=
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
-
9
=
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
14
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+4
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
-
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
-
9
-
9
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
-
9
-
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
-
9
=
9
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
-
9
=
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

6
ENERGY
74
38
2
8
ENERGIES
82
46
1

 

 

5
FORCE
47
29
2
6
FORCES
66
30
3

 

 

-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
1
-
6
-
-
-
-
-
-
9
-
-
+
=
22
2+2
=
4
=
4
=
4
-
-
-
15
-
-
-
19
-
15
-
-
-
-
-
-
9
-
-
+
=
58
5+8
=
13
1+3
4
=
4
-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
6
-
9
3
5
-
-
-
6
-
7
9
1
4
-
2
7
+
=
59
5+9
=
14
1+4
5
=
5
-
-
6
-
18
3
5
-
-
-
6
-
7
18
1
22
-
20
25
+
=
131
1+3+1
=
5
=
5
=
5
-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
6
15
18
3
5
19
-
15
6
-
7
18
1
22
9
20
25
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
6
6
9
3
5
1
-
6
6
-
7
9
1
4
9
2
7
+
=
81
8+1
=
9
=
9
=
9
-
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-``
-
6
6
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
8
15
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
37
-
-
15
-
81
-
36
-
1+5
-
-
9
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
3+7
-
-
1+5
-
8+1
-
3+6
8
6
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
10
-
-
6
-
9
-
9
-
-
6
6
9
3
5
1
-
6
6
-
7
9
1
4
9
2
7
-
-
1+0
-
-
-
-
-
-
-
8
6
F
O
R
C
E
S
-
O
F
-
G
R
A
V
I
T
Y
-
-
1
-
-
6
-
9
-
9

 

 

4
HEAT
34
16
7
5
LIGHT
56
29
2
9
Add to Reduce
90
45
9
-
Reduce to Deduce
9+0
4+5
-
9
Essence of Number
9
9
9

 

 

4
MIND
40
22
4
6
MATTER
77
23
5
10
Add to Reduce
117
45
9
1+0
Reduce to Deduce
1+1+7
4+5
-
1
Essence of Number
9
9
9

 

 

--
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
-
6
1
-
6
-
-
+
=
14
1+4
=
5
-
5
-
5
-
-
19
-
-
-
-
-
15
19
-
15
-
-
+
=
68
6+8
=
14
1+4
5
-
5
--
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
-
7
5
3
2
9
-
-
3
-
7
5
+
=
41
4+1
=
5
-
5
-
5
-
-
-
16
5
3
20
18
-
-
3
-
16
5
+
=
86
8+6
=
14
1+4
5
-
5
--
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
19
16
5
3
20
18
15
19
3
15
16
5
+
+
154
1+5+4
=
10
1+0
1
-
1
-
-
1
7
5
3
2
9
6
1
3
6
7
5
+
+
55
5+5
=
10
1+0
1
-
1
-
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
--
1
-
--
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
--
1
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
-
-
3
-
-
-
-
3
-
-
-
-
-
3
occurs
x
2
=
6
-
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
12
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
33
-
-
12
-
55
-
28
1+2
1+2
-
-
-
-
-
9
-
-
-
-
-
-
-
-
3+3
-
-
1+2
-
5+5
-
2+8
3
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
10
-
10
-
-
1
7
5
3
2
9
6
1
3
6
7
5
-
-
-
-
-
-
-
1+0
-
1+0
3
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
1
-
1

 

 

12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
-
6
1
-
6
-
-
+
=
14
1+4
=
5
-
5
-
5
-
19
-
-
-
-
-
15
19
-
15
-
-
+
=
68
6+8
=
14
1+4
5
-
5
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
7
5
3
2
9
-
-
3
-
7
5
+
=
41
4+1
=
5
-
5
-
5
-
-
16
5
3
20
18
-
-
3
-
16
5
+
=
86
8+6
=
14
1+4
5
-
5
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
19
16
5
3
20
18
15
19
3
15
16
5
+
+
154
1+5+4
=
10
1+0
1
-
1
-
1
7
5
3
2
9
6
1
3
6
7
5
+
+
55
5+5
=
10
1+0
1
-
1
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
--
1
-
--
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
--
1
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
-
3
-
-
-
-
3
-
-
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
6
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
12
S
P
E
C
T
R
O
S
C
O
P
E
-
-
33
-
-
12
-
55
-
28
1+2
-
-
-
-
-
9
-
-
-
-
-
-
-
-
3+3
-
-
1+2
-
5+5
-
2+8
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
10
-
10
-
1
7
5
3
2
9
6
1
3
6
7
5
-
-
-
-
-
-
-
1+0
-
1+0
3
S
P
E
C
T
R
O
S
C
O
P
E
-
-
6
-
-
3
-
1
-
1

 

 

-
SPECTRUM
-
-
-
5
SPECT
63
18
9
1
R
18
9
9
2
UM
34
7
7
8
SPECTRUM
115
34
9
-
-
1+1+5
3+4
-
8
SPECTRUM
7
7
7

 

 

--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
-
=
1
-
1
-
-
19
-
-
-
-
-
-
-
+
=
19
1+9
=
10
1+0
1
--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
--
-
-
-
-
-
7
5
3
2
9
3
4
+
=
33
3+3
=
6
-
6
-
-
-
16
5
3
20
18
21
13
+
=
96
96
=
15
1+5
6
--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
--
-
-
-
-
19
16
5
3
20
18
21
13
+
+
115
1+1+5
=
7
-
7
-
-
1
7
5
3
2
9
3
4
+
+
34
3+4
=
7
-
7
-
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
-
--
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
-
3
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
5
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
14
8
S
P
E
C
T
R
U
M
-
-
33
-
-
8
-
34
1+4
-
1
7
5
3
2
9
-
-
-
-
3+3
-
-
-
-
3+4
5
8
S
P
E
C
T
R
U
M
-
-
6
-
-
3
-
7
-
-
1
7
5
3
2
9
3
4
-
-
-
-
-
-
-
-
5
8
S
P
E
C
T
R
U
M
-
-
6
-
-
3
-
7

 

 

--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
-
=
1
-
1
-
-
19
-
-
-
-
-
-
-
+
=
19
1+9
=
10
1+0
1
--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
--
-
-
-
-
-
7
5
3
2
9
3
4
+
=
33
3+3
=
6
-
6
-
-
-
16
5
3
20
18
21
13
+
=
96
96
=
15
1+5
6
--
8
S
P
E
C
T
R
U
M
-
-
-
-
-
--
-
-
-
-
19
16
5
3
20
18
21
13
+
+
115
1+1+5
=
7
-
7
-
-
1
7
5
3
2
9
3
4
+
+
34
3+4
=
7
-
7
-
8
S
P
E
C
T
R
U
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
-
--
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
-
3
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
5
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
14
8
S
P
E
C
T
R
U
M
-
-
33
-
-
8
-
34
1+4
-
1
7
5
3
2
9
-
-
-
-
3+3
-
-
-
-
3+4
5
8
S
P
E
C
T
R
U
M
-
-
6
-
-
3
-
7
-
-
1
7
5
3
2
9
3
4
-
-
-
-
-
-
-
-
5
8
S
P
E
C
T
R
U
M
-
-
6
-
-
3
-
7

 

 

7
RAINBOW
82
37
1
6
COLOUR
84
30
3

 

 

7
RAINBOW
82
37
1
7
COLOURS
103
31
4

 

 

8
SPECTRUM
115
34
7
2
OF
21
12
3
3
THE
33
15
6
7
RAINBOW
82
37
1

 

 

3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2

 

 

3
THE
33
15
6
7
RAINBOW
82
37
1
8
SPECTRUM
115
34
7
2
OF
21
12
3
6
COLOUR
84
30
3

 

 

8
COLOURED
-
-
-
-
C+O
18
9
9
9
L+O
27
9
9
9
U
21
3
3
-
R
18
9
9
9
E+D
9
9
9
8
COLOURED
-
-
-

 

 

6
COLOUR
84
30
3
3
RED
27
18
9

 

 

Kaleidoscope - Wikipedia, the free encyclopedia
A kaleidoscope is a tube of mirrors containing loose colored beads, pebbles or other small colored objects. The viewer looks in one end and light enters the ... en.wikipedia.org/wiki/Kaleidoscope

Giant kaleidoscope, San Diego harbor. A kaleidoscope is a tube of mirrors containing loose colored beads, pebbles or other small colored objects. The viewer looks in one end and light enters the other end, reflecting off the mirrors. Typically there are two rectangular lengthwise mirrors. Setting of the mirrors at 45° creates eight duplicate images of the objects, four at 60°, and four at 90°. As the tube is rotated, the tumbling of the colored objects presents the viewer with varying colors and patterns. Any arbitrary pattern of objects shows up as a beautiful symmetric pattern because of the reflections in the mirrors. A two-mirror model yields a pattern or patterns isolated against a solid black background, while a three-mirror (closed triangle) model yields a pattern that fills the entire field. For a 2D symmetry group, a kaleidoscopic point is a point of intersection of two or more lines of reflection symmetry. In the case of a discrete group the angle between consecutive lines is 180°/n for an integer n≥2. At this point there are n lines of reflection symmetry, and the point is a center of n-fold rotational symmetry. See also symmetry combinations. Modern kaleidoscope are made of brass tubes, stained glass, wood, steel, gourds and most any other material an artist can sculpt or manipulate. The part of the kaleidoscope which holds objects to be viewed is called an object chamber or cell. Object cells may contain almost any material. Sometimes the object cell is filled with liquid so the items float and move through the object cell with slight movement from the person viewing.

Contents [hide]
1 Etymology
2 History
3 Background
4 See also
5 References
6 External links

[edit] Etymology

A toy kaleidoscope tube. First attested 1817 in English, the word "kaleidoscope" derives from the Greek καλός (kalos), "beautiful"[1] + είδος (eidos), "shape"[2] + σκοπέω (scopeο), "to look at, to examine"[3][4].

More informatively: "looking at beautiful forms" (PB May 2009)

[edit] History

Patterns as seen through a kaleidoscope tube. Known to the ancient Greeks, it was reinvented by Sir David Brewster in 1816 while conducting experiments on light polarization; Brewster patented it in 1817. His initial design was a tube with pairs of mirrors at one end, and pairs of translucent disks at the other, and beads between the two. Initially intended as a science tool, the kaleidoscope was quickly copied as a toy. Brewster believed he would make money from his popular invention; however, a fault in the wording of his patent allowed others to copy his invention.[citation needed]

In America, Charles Bush popularized the kaleidoscope. Today, these early products often sell for over $1,000. Cozy Baker collected kaleidoscopes and wrote books about a few of the artists who were making them in the 1970s through 2000. Baker is credited with energizing a renaissance in kaleidoscope-making in America. In 1999 a short lived magazine dedicated to kaleidoscopes called Kaleidoscope Review was published covering artists, collectors, dealers, events, and how-to articles. This magazine was created and edited by Brett Bensley, at that time a well known kaleidoscope artist and resource on kaleidoscope information.

Craft galleries often carry a few, while others specialize in them and carry dozens of different types from different artists and craftspeople.

Kaleidoscopes are related to hyperbolic geometry.

[edit] Background
For some background on the geometry of the kaleidoscope, see Reflection group.

[edit] See also
Teleidoscope
Kaleidoplex
Fractals

[edit] References
1.^ Kalos, Henry George Liddell, Robert Scott, "A Greek-English Lexicon", at Perseus
2.^ Eidos, Henry George Liddell, Robert Scott, "A Greek-English Lexicon", at Perseus
3.^ Skopeo, Henry George Liddell, Robert Scott, "A Greek-English Lexicon", at Perseus
4.^ Online Etymology Dictionary

[edit] External links
Media related to Kaleidoscopes at Wikimedia Commons
[1] spectacular kaleidoscope on youtube
[2] another spectacular kaleidoscope on youtube (links to more)
12 in HU, moving interactive kaleidoscope
Literary and Philosophical Remarks.
Kaleidoscope History
How to make a Kaleidoscope
Kaleidoscope Mirror Designs
Infography about Kaleidoscopes
Digital kaleidoscope gallery
Customizable interactive animated Kaleidoscope application for iPhone

 

 

-
12
K
A
L
E
I
D
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
9
-
6
1
-
6
-
-
+
=
22
2+2
=
4
-
4
=
4
-
-
-
-
-
-
9
-
15
19
-
15
-
-
+
=
58
5+8
=
13
1+3
4
=
4
-
12
K
A
L
E
I
D
O
S
C
O
P
E
-
-
-
-
-
-
-
-
-
-
-
-
2
1
3
5
-
4
-
-
3
-
7
5
+
=
30
3+9
=
12
1+2
3
=
3
-
-
11
1
12
5
-
4
-
-
3
-
16
5
+
=
57
5+7
=
12
1+2
3
=
3
-
12
K
A
L
E
I
D
O
S
C
O
P
E
-
-
-
-
-
--
-
-
-
-
-
-
11
1
12
5
9
4
15
19
3
15
16
5
+
=
115
1+1+5
=
7
-
7
=
7
-
-
2
1
3
5
9
4
6
1
3
6
7
5
+
=
52
5+2
=
7
-
7
=
7
-
12
K
A
L
E
I
D
O
S
C
O
P
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
3
-
-
-
-
-
3
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
8
12
K
A
L
E
I
D
O
S
C
O
P
E
-
-
37
-
-
12
-
52
-
34
-
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+7
-
-
1+2
-
5+2
-
3+4
8
3
K
A
L
E
I
D
O
S
C
O
P
E
-
-
10
-
-
3
-
7
-
7
-
-
2
1
3
5
9
4
6
1
3
6
7
5
-
-
1+0
-
-
-
-
-
-
-
8
3
K
A
L
E
I
D
O
S
C
O
P
E
-
-
1
-
-
3
-
7
-
7

 

 

Q
=
8
7
QUANTUM
107
26
8
T
=
2
6
THEORY
91
37
1
-
-
10
13
First Total
198
63
9
-
-
1+0
1+3
Add to Reduce
1+9+8
6+3
-
-
-
1
4
Second Total
18
9
9
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
1
4
Essence of Number
9
9
9

 

 

-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
8
-
6
-
-
+
=
19
1+9
=
10
1+0
1
=
1
-
-
-
-
-
14
-
-
-
-
-
8
-
15
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
2
-
5
-
9
7
+
=
44
4+4
=
8
-
8
=
8
-
-
17
21
1
-
20
21
13
-
20
-
5
-
18
25
+
=
161
1+6+1
=
8
-
8
=
8
-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
20
8
5
15
18
25
+
+
198
1+9+8
=
18
1+8
9
=
9
-
-
8
3
1
5
2
3
4
-
2
8
5
6
9
7
+
+
63
6+3
=
9
-
9
=
9
-
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
``-
-
-
-
-
2
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
3
``-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
-
6
4
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
8
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
45
13
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
45
-
-
13
-
63
-
45
4+5
1+3
-
-
-
-
-
-
-``
-
-
-
-
-
9
-
-
-
4+5
-
-
1+3
-
6+3
-
4+5
9
4
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
9
-
-
4
-
9
-
9
-
-
8
3
1
5
2
3
4
-
2
8
5
6
9
7
-
-
-
-
-
-
-
-
-
-
9
4
Q
U
A
N
T
U
M
-
T
H
E
O
R
Y
-
-
9
-
-
4
-
9
-
9

 

 

-
7
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
9
-
5
-
-
+
=
15
1+5
=
6
-
6
=
6
-
-
19
-
9
-
14
-
-
+
=
42
4+2
=
6
-
6
-
6
-
7
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
5
-
3
5
+
=
16
1+6
=
7
=
7
-
7
-
7
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
19
3
9
5
14
3
5
+
=
58
5+8
=
13
1+3
4
-
4
-
-
1
3
9
5
5
3
5
+
=
31
3+1
=
4
=
4
-
4
-
7
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
5
5
-
5
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
27
7
S
C
I
E
N
C
E
-
-
18
-
-
7
-
31
-
22
2+7
-
-
-
9
-
-
-
-
-
-
1+8
-
-
-
-
3+1
-
2+2
9
7
S
C
I
E
N
C
E
-
-
9
-
-
7
-
4
-
4

 

 

7
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
1
-
9
-
5
-
-
+
=
15
1+5
=
6
-
6
=
6
-
19
-
9
-
14
-
-
+
=
42
4+2
=
6
-
6
-
6
7
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
3
-
5
-
3
5
+
=
16
1+6
=
7
=
7
-
7
7
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
19
3
9
5
14
3
5
+
=
58
5+8
=
13
1+3
4
-
4
-
1
3
9
5
5
3
5
+
=
31
3+1
=
4
=
4
-
4
7
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
5
5
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
7
S
C
I
E
N
C
E
-
-
18
-
-
7
-
31
-
22
-
-
-
9
-
-
-
-
-
-
1+8
-
-
-
-
3+1
-
2+2
7
S
C
I
E
N
C
E
-
-
9
-
-
7
-
4
-
4

 

 

7
SCIENCE
58
31
4
10
CONSCIENCE
90
36
9

 

 

-
7
S
I
L
I
C
O
N
-
-
-
-
-
--
-
-
-
-
-
-
1
9
-
9
-
6
5
+
=
30
3+0
=
3
=
3
=
3
-
-
19
9
-
9
-
15
14
+
=
66
6+6
=
12
1+2
3
-
3
-
7
S
I
L
I
C
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
-
-
+
=
6
-
=
6
=
6
-
6
-
-
-
-
12
-
3
-
-
+
=
15
1+5
=
6
=
6
-
6
-
7
S
I
L
I
C
O
N
-
-
-
-
-
-
-
-
-
-
-
-
19
9
12
9
3
15
14
+
=
81
8+1
=
9
=
9
-
9
-
-
1
9
3
9
3
6
5
+
=
36
3+6
=
9
=
9
-
9
-
7
S
I
L
I
C
O
N
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
21
7
S
I
L
I
C
O
N
-
-
24
-
-
7
-
36
-
27
2+1
-
-
9
-
9
-
-
-
-
-
2+4
-
-
-
-
3+6
-
2+7
3
7
S
I
L
I
C
O
N
-
-
6
-
-
7
-
9
-
9

 

 

7
S
I
L
I
C
O
N
-
-
-
-
-
--
-
-
-
-
-
1
9
-
9
-
6
5
+
=
30
3+0
=
3
=
3
=
3
-
19
9
-
9
-
15
14
+
=
66
6+6
=
12
1+2
3
-
3
7
S
I
L
I
C
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
-
-
+
=
6
-
=
6
=
6
-
6
-
-
-
12
-
3
-
-
+
=
15
1+5
=
6
=
6
-
6
7
S
I
L
I
C
O
N
-
-
-
-
-
-
-
-
-
-
-
19
9
12
9
3
15
14
+
=
81
8+1
=
9
=
9
-
9
-
1
9
3
9
3
6
5
+
=
36
3+6
=
9
=
9
-
9
7
S
I
L
I
C
O
N
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
7
S
I
L
I
C
O
N
-
-
24
-
-
7
-
36
-
27
-
-
9
-
9
-
-
-
-
-
2+4
-
-
-
-
3+6
-
2+7
7
S
I
L
I
C
O
N
-
-
6
-
-
7
-
9
-
9

 

 

-
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
9
-
-
+
=
14
1+4
=
5
=
5
-
5
-
`-
-
-
-
14
9
-
-
+
=
23
2+3
=
5
=
5
-
5
-
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
3
9
1
-
-
3
4
+
=
20
2+0
=
2
=
2
-
2
-
-
21
18
1
-
-
21
13
+
=
74
7+4
=
11
1+1
2
-
2
-
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
`-
21
18
1
14
9
21
13
+
=
97
9+7
=
16
1+6
7
-
7
-
-
3
9
1
5
9
3
4
+
=
34
3+4
=
7
=
7
-
7
-
7
U
R
A
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
1
1
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
4
-
-
4
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
4
-
-
4
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
-
-
5
-
-
5
occurs
x
1
=
5
=
5
6
-
-
4
-
-
4
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
4
-
-
4
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
4
-
-
4
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
9
-
-
9
9
9
-
-
9
occurs
x
2
=
18
1+8
9
23
7
U
R
A
N
I
U
M
-
-
22
-
-
7
-
33
-
24
2+3
-
9
9
-
9
9
-
-
-
2+2
-
-
-
-
3+3
-
2+4
5
7
U
R
A
N
I
U
M
-
-
4
-
-
7
-
6
-
6

 

 

7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
9
-
-
+
=
14
1+4
=
5
=
5
-
5
`-
-
-
-
14
9
-
-
+
=
23
2+3
=
5
=
5
-
5
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
3
9
1
-
-
3
4
+
=
20
2+0
=
2
=
2
-
2
-
21
18
1
-
-
21
13
+
=
74
7+4
=
11
1+1
2
-
2
7
U
R
A
N
I
U
M
-
-
-
-
-
--
-
-
-
-
`-
21
18
1
14
9
21
13
+
=
97
9+7
=
16
1+6
7
-
7
-
3
9
1
5
9
3
4
+
=
34
3+4
=
7
=
7
-
7
7
U
R
A
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
1
1
1
-
-
-
-
1
occurs
x
1
=
1
=
1
-
3
4
-
-
4
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
5
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
9
9
-
-
9
9
9
-
-
9
occurs
x
2
=
18
1+8
9
7
U
R
A
N
I
U
M
-
-
22
-
-
7
-
33
-
24
-
9
9
-
9
9
-
-
-
2+2
-
-
-
-
3+3
-
2+4
7
U
R
A
N
I
U
M
-
-
4
-
-
7
-
6
-
6

 

 

--
7
H
A
F
N
I
U
M
-
-
-
-
-
--
-
-
-
-
8
-
-
5
9
-
-
+
=
22
2+2
=
4
=
4
-
-
8
-
-
14
9
-
-
+
=
31
3+1
=
4
=
4
--
7
H
A
F
N
I
U
M
-
-
-
-
-
--
-
-
-
-
-
1
6
-
-
3
4
+
=
14
1+4
=
5
=
5
-
-
-
1
6
-
-
21
13
+
=
41
4+1
=
5
=
5
--
7
H
A
F
N
I
U
M
-
-
-
-
-
--
-
-
-
-
8
1
6
14
9
21
13
+
=
72
7+2
=
9
=
9
-
-
8
1
6
5
9
3
4
+
=
36
3+6
=
9
=
9
-
7
H
A
F
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
6
-
-
-
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
9
7
H
A
F
N
I
U
M
-
-
36
-
-
7
-
36
-
-
-
-
-
-
9
-
-
-
-
3+6
-
-
-
-
3+6
9
7
H
A
F
N
I
U
M
-
-
9
-
-
7
-
9

 

 

10
SUBJECTIVE
116
35
8
9
OBJECTIVE
91
37
1
19
Add to Reduce
207
72
9
10
Reduce to Deduce
2+0+7
7+2
-
1
Essence of Number
9
9
9

 

 

S
=
1
7
SUBJECT
80
17
8
O
=
6
6
OBJECT
55
19
1
-
-
7
13
-
135
36
9
-
-
-
1+3
-
1+3+5
3+6
-
Q
-
7
4
-
9
9
9

 

 

-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
-
-
-
6
-
-
-
-
-
+
=
7
-
=
7
=
7
-
-
-
-
19
-
-
-
-
-
-
-
15
-
-
-
-
-
+
=
34
3+4
=
7
=
7
-
-
-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
-
3
2
1
5
3
2
-
-
2
1
5
3
2
+
=
29
2+9
=
11
1+1
2
-
-
-
-
-
21
2
10
5
3
20
-
-
2
10
5
3
20
+
=
20
2+0
=
2
=
2
-
-
-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
19
21
20
10
5
3
20
-
15
2
10
5
3
20
+
=
135
1+3+5
=
9
=
9
-
-
-
-
1
3
2
1
5
3
2
-
6
2
1
5
3
2
+
=
36
3+6
=
9
=
9
-
-
-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
``-
-
-
2
-
-
-
2
-`
-
2
-
-
-
2
-
-
2
occurs
x
4
=
8
=
8
-
-
-
3
-
-
-
3
-
-
-
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
28
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
17
-
-
13
-
36
-
27
2+8
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+7
-
-
1+3
-
3+6
-
2+7
10
4
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
8
-
-
4
-
9
-
9
1+0
-
1
3
2
1
5
3
2
-
6
2
1
5
3
2
-
-
-
-
-
-
-
-
-
-
1
4
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
8
-
-
4
-
9
-
9

 

 

13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
-
-
-
6
-
-
-
-
-
+
=
7
-
=
7
=
7
-
-
-
19
-
-
-
-
-
-
-
15
-
-
-
-
-
+
=
34
3+4
=
7
=
7
-
-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
3
2
1
5
3
2
-
-
2
1
5
3
2
+
=
29
2+9
=
11
1+1
2
-
-
-
-
21
2
10
5
3
20
-
-
2
10
5
3
20
+
=
20
2+0
=
2
=
2
-
-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
--
-
-
-
-
-
19
21
20
10
5
3
20
-
15
2
10
5
3
20
+
=
135
1+3+5
=
9
=
9
-
-
-
1
3
2
1
5
3
2
-
6
2
1
5
3
2
+
=
36
3+6
=
9
=
9
-
-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
2
-
-
-
2
-`
-
2
-
-
-
2
-
-
2
occurs
x
4
=
8
=
8
-
-
3
-
-
-
3
-
-
-
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
13
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
17
-
-
13
-
36
-
27
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+7
-
-
1+3
-
3+6
-
2+7
4
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
8
-
-
4
-
9
-
9
-
1
3
2
1
5
3
2
-
6
2
1
5
3
2
-
-
-
-
-
-
-
-
-
-
4
S
U
B
J
E
C
T
-
O
B
J
E
C
T
-
-
8
-
-
4
-
9
-
9

 

 

 

-
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
-
-
9
-
-
-
6
-
-
-
-
-
9
-
-
+
=
25
2+5
=
7
=
7
-
7
-
-
19
-
-
-
-
-
-
9
-
-
-
15
-
-
-
-
-
9
-
-
+
=
52
5+2
=
7
=
7
-
7
-
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
--
-
-
-
-
-
-
-
3
2
1
5
3
2
-
4
5
-
-
2
1
5
3
2
-
4
5
+
=
47
4+7
=
11
1+1
2
-
2
-
-
-
21
2
10
5
3
20
-
22
5
-
-
2
10
5
3
20
-
22
5
+
=
155
1+5+5
=
11
1+1
2
-
2
-
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
--
-
-
-
-
-
-
19
21
20
10
5
3
20
9
22
5
-
15
2
10
5
3
20
9
22
5
+
=
207
2+0+7
=
9
=
9
-
9
-
-
1
3
2
1
5
3
2
9
4
5
-
6
2
1
5
3
2
9
4
5
+
=
72
7+2
=
9
=
9
-
9
-
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
``-
-
-
2
-
-
-
2
-
-
-
-`
-
2
-
-
-
2
-
-
-
-
-
2
occurs
x
4
=
8
=
8
-
-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
15
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
30
-
-
19
-
72
-
45
1+5
1+9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3+0
-
-
10
-
7+2
-
4+5
6
10
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
3
-
-
1
-
9
-
9
-
1+0
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
--
-
-
-
-
6
1
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
3
-
-
1
-
9
-
9

 

 

19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
-
-
9
-
-
-
6
-
-
-
-
-
9
-
-
+
=
25
2+5
=
7
=
7
-
7
-
19
-
-
-
-
-
-
9
-
-
-
15
-
-
-
-
-
9
-
-
+
=
52
5+2
=
7
=
7
-
7
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
--
-
-
-
-
-
-
3
2
1
5
3
2
-
4
5
-
-
2
1
5
3
2
-
4
5
+
=
47
4+7
=
11
1+1
2
-
2
-
-
21
2
10
5
3
20
-
22
5
-
-
2
10
5
3
20
-
22
5
+
=
155
1+5+5
=
11
1+1
2
-
2
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
--
-
-
-
-
-
19
21
20
10
5
3
20
9
22
5
-
15
2
10
5
3
20
9
22
5
+
=
207
2+0+7
=
9
=
9
-
9
-
1
3
2
1
5
3
2
9
4
5
-
6
2
1
5
3
2
9
4
5
+
=
72
7+2
=
9
=
9
-
9
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
2
-
-
-
2
-
-
-
-`
-
2
-
-
-
2
-
-
-
-
-
2
occurs
x
4
=
8
=
8
-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
19
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
30
-
-
19
-
72
-
45
1+9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3+0
-
-
10
-
7+2
-
4+5
10
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
3
-
-
1
-
9
-
9
1+0
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
--
-
-
-
-
1
S
U
B
J
E
C
T
I
V
E
-
O
B
J
E
C
T
I
V
E
-
-
3
-
-
1
-
9
-
9

 

 

7
SUBJECT
80
17
8
8
SUBJECTS
99
18
9

 

 

6
OBJECT
55
19
1
7
OBJECTS
74
20
2

 

 

-
6
M
I
R
R
O
R
-
-
-
-
-
--
-
-
-
-
-
-
-
9
-
-
6
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
9
-
-
15
-
+
=
24
2+4
=
6
=
6
=
6
-
6
M
I
R
R
O
R
-
-
-
-
-
--
-
-
-
-
-
-
4
-
9
9
-
9
+
=
31
3+1
=
4
=
4
=
4
-
-
13
-
18
18
-
18
+
=
67
6+7
=
13
1+3
4
=
4
-
6
M
I
R
R
O
R
-
-
-
-
-
--
-
-
-
-
-
-
13
9
18
18
15
18
+
=
91
9+1
=
10
1+0
1
=
1
-
-
4
9
9
9
6
9
+
=
46
4+6
=
10
1+0
1
=
1
-
6
M
I
R
R
O
R
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
--
--
--
-
--
1
-
-
-
-
-
-
-
2
-
-
-
-
--
--
--
-
--
2
-
-
-
-
-
-
-
3
-
-
-
-
--
--
--
-
--
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
--
--
--
-
--
5
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
--
--
--
-
--
7
-
-
-
-
-
-
-
8
-
-
-
-
--
--
--
-
--
8
-
-
-
-
-
-
-
-
-
-
9
9
9
-
9
-
-
9
occurs
x
4
=
36
3+6
9
26
6
M
I
R
R
O
R
-
-
19
-
-
6
-
46
-
19
2+6
-
-
9
9
9
-
9
-
-
1+9
-
-
-
-
4+6
-
1+9
8
6
M
I
R
R
O
R
-
-
10
-
-
6
-
10
-
10
-
-
4
9
9
9
6
9
-
-
1+0
-
-
-
-
1+0
-
1+0
8
6
M
I
R
R
O
R
-
-
1
-
-
6
-
1
-
1

 

 

6
M
I
R
R
O
R
-
-
-
-
-
--
-
-
-
-
-
-
9
-
-
6
-
+
=
15
1+5
=
6
=
6
=
6
-
-
9
-
-
15
-
+
=
24
2+4
=
6
=
6
=
6
6
M
I
R
R
O
R
-
-
-
-
-
--
-
-
-
-
-
4
-
9
9
-
9
+
=
31
3+1
=
4
=
4
=
4
-
13
-
18
18
-
18
+
=
67
6+7
=
13
1+3
4
=
4
6
M
I
R
R
O
R
-
-
-
-
-
--
-
-
-
-
-
13
9
18
18
15
18
+
=
91
9+1
=
10
1+0
1
=
1
-
4
9
9
9
6
9
+
=
46
4+6
=
10
1+0
1
=
1
6
M
I
R
R
O
R
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
-
-
9
9
9
-
9
-
-
9
occurs
x
4
=
36
3+6
9
6
M
I
R
R
O
R
-
-
19
-
-
6
-
46
-
19
-
-
9
9
9
-
9
-
-
1+9
-
-
-
-
4+6
-
1+9
6
M
I
R
R
O
R
-
-
10
-
-
6
-
10
-
10
-
4
9
9
9
6
9
-
-
1+0
-
-
-
-
1+0
-
1+0
6
M
I
R
R
O
R
-
-
1
-
-
6
-
1
-
1

 

 

-
11
M
I
R
R
O
R
--
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
-
-
-
9
-
-
6
-
--
9
-
-
-
-
+
=
24
2+4
=
6
=
6
=
6
-
-
-
9
-
-
15
-
--
9
-
-
-
-
+
=
33
3+3
=
6
=
6
=
6
-
11
M
I
R
R
O
R
-
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
-
-
4
-
9
9
-
9
--
-
4
1
7
5
+
=
48
4+8
=
12
1+2
3
=
3
-
-
13
-
18
18
-
18
-
-
13
1
7
5
+
=
93
9+3
=
12
1+2
3
=
3
-
11
M
I
R
R
O
R
--
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
-
-
13
9
18
18
15
18
--
9
13
1
7
5
+
=
126
1+2+6
=
9
=
9
=
9
-
-
4
9
9
9
6
9
-
9
4
1
7
5
+
=
72
7+2
=
9
=
9
=
9
-
11
M
I
R
R
O
R
-
I
M
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
--
--
-
-
-
1
-
-
-
--
1
occurs
x
1
=
1
=
1
2
-
-
-
-
--
--
--
-
-
-
-
-
-
-
--
2
-
-
-
-
-
-
-
3
-
-
-
-
--
--
--
-
-
-
-
-
-
-
--
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
--
--
--
-
-
-
-
-
5
-
--
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
--
--
--
-
-
-
-
7
-
-
--
7
occurs
x
1
=
7
=
7
8
-
-
-
-
--
--
--
-
-
-
-
-
-
-
--
8
-
-
-
-
-
-
-
-
-
-
9
9
9
-
9
-
9
-
-
-
-
-
-
9
occurs
x
5
=
45
3+6
9
13
11
M
I
R
R
O
R
-
I
M
A
G
E
-
-
32
-
-
11
-
72
-
36
1+3
1+1
-
9
9
9
-
9
-
9
-
-
-
-
-
-
3+2
-
-
1+1
-
7+2
-
3+6
4
2
M
I
R
R
O
R
-
I
M
A
G
E
-
-
5
-
-
2
-
9
-
9
-
-
4
9
9
9
6
9
-
9
4
1
7
5
-
-
-
-
-
-
-
-
-
-
4
2
M
I
R
R
O
R
-
I
M
A
G
E
-
-
5
-
-
2
-
9
-
9

 

 

11
M
I
R
R
O
R
--
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
-
-
9
-
-
6
-
--
9
-
-
-
-
+
=
24
2+4
=
6
=
6
=
6
-
-
9
-
-
15
-
--
9
-
-
-
-
+
=
33
3+3
=
6
=
6
=
6
11
M
I
R
R
O
R
-
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
-
4
-
9
9
-
9
--
-
4
1
7
5
+
=
48
4+8
=
12
1+2
3
=
3
-
13
-
18
18
-
18
-
-
13
1
7
5
+
=
93
9+3
=
12
1+2
3
=
3
11
M
I
R
R
O
R
--
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
-
13
9
18
18
15
18
--
9
13
1
7
5
+
=
126
1+2+6
=
9
=
9
=
9
-
4
9
9
9
6
9
-
9
4
1
7
5
+
=
72
7+2
=
9
=
9
=
9
11
M
I
R
R
O
R
-
I
M
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
--
--
-
-
-
1
-
-
-
--
1
occurs
x
1
=
1
=
1
-
4
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
--
--
--
-
-
-
-
-
5
-
--
5
occurs
x
1
=
5
=
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
--
--
--
-
-
-
-
7
-
-
--
7
occurs
x
1
=
7
=
7
-
-
9
9
9
-
9
-
9
-
-
-
-
-
-
9
occurs
x
5
=
45
3+6
9
11
M
I
R
R
O
R
-
I
M
A
G
E
-
-
32
-
-
11
-
72
-
36
1+1
-
9
9
9
-
9
-
9
-
-
-
-
-
-
3+2
-
-
1+1
-
7+2
-
3+6
2
M
I
R
R
O
R
-
I
M
A
G
E
-
-
5
-
-
2
-
9
-
9
-
4
9
9
9
6
9
-
9
4
1
7
5
-
-
-
-
-
-
-
-
-
-
2
M
I
R
R
O
R
-
I
M
A
G
E
-
-
5
-
-
2
-
9
-
9

 

 

-`
5
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
9
-
-
-
-
+
=
10
1+0
=
1
=
1
-
-
9
-
-
-
-
+
=
28
2+8
=
10
1+0
1
-`
5
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
-
4
1
7
5
+
=
17
1+7
=
8
=
8
-
-
-
13
1
7
5
+
=
26
2+6
=
8
=
8
-`
5
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
9
13
1
7
5
+
=
35
3+5
=
8
=
8
-
-
9
4
1
7
5
+
=
26
2+6
=
8
=
8
-
5
I
M
A
G
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
4
-
-
-
=
-
4
occurs
x
1
=
4
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
7
-
=
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
=
-
9
occurs
x
1
=
9
19
5
I
M
A
G
E
-
-
26
-
-
5
-
26
1+9
-
9
-
-
-
-
-
-
2+6
-
-
-
-
2+6
10
5
I
M
A
G
E
-
-
8
-
-
5
-
8
1+0
-
9
4
1
7
5
-
-
-
-
-
--
-
-
9
5
I
M
A
G
E
-
-
8
-
-
5
-
8

 

 

5
I
M
A
G
E
-
-
-
-
-
--
-
-
-
9
-
-
-
-
+
=
10
1+0
=
1
=
1
-
9
-
-
-
-
+
=
28
2+8
=
10
1+0
1
5
I
M
A
G
E
-
-
-
-
-
--
-
-
-
-
4
1
7
5
+
=
17
1+7
=
8
=
8
-
-
13
1
7
5
+
=
26
2+6
=
8
=
8
5
I
M
A
G
E
-
-
-
-
-
--
-
-
-
9
13
1
7
5
+
=
35
3+5
=
8
=
8
-
9
4
1
7
5
+
=
26
2+6
=
8
=
8
5
I
M
A
G
E
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
-
-
4
-
-
-
=
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
-
-
-
7
-
=
-
7
occurs
x
1
=
7
-
9
-
-
-
-
=
-
9
occurs
x
1
=
9
5
I
M
A
G
E
-
-
26
-
-
5
-
26
-
9
-
-
-
-
-
-
2+6
-
-
-
-
2+6
5
I
M
A
G
E
-
-
8
-
-
5
-
8
-
9
4
1
7
5
-
-
-
-
-
--
-
-
5
I
M
A
G
E
-
-
8
-
-
5
-
8

 

 

-`
6
I
M
A
G
E
S
-
-
-
-
-
--
-
-
-
-
9
-
-
-
-
1
+
=
10
1+0
=
1
=
1
-
-
9
-
-
-
-
19
+
=
28
2+8
=
10
1+0
1
-`
6
I
M
A
G
E
S
-
-
-
-
-
--
-
-
-
-
-
4
1
7
5
-
+
=
17
1+7
=
8
=
8
-
-
-
13
1
7
5
-
+
=
26
2+6
=
8
=
8
-`
6
I
M
A
G
E
S
-
-
-
-
-
--
-
-
-
-
9
13
1
7
5
19
+
=
54
5+4
=
9
=
9
-
-
9
4
1
7
5
1
+
=
27
2+7
=
9
=
9
-
6
I
M
A
G
E
S
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
4
-
-
-
-
=
-
4
occurs
x
1
=
4
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
7
-
-
=
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
=
-
9
occurs
x
1
=
9
19
6
I
M
A
G
E
S
-
-
26
-
-
6
-
27
1+9
-
9
-
-
-
-
-
-
-
2+6
-
-
-
-
2+7
10
6
I
M
A
G
E
S
-
-
8
-
-
6
-
9
1+0
-
9
4
1
7
5
1
-
-
-
-
-
--
-
-
9
6
I
M
A
G
E
S
-
-
8
-
-
6
-
9

 

 

6
I
M
A
G
E
S
-
-
-
-
-
--
-
-
-
9
-
-
-
-
1
+
=
10
1+0
=
1
=
1
-
9
-
-
-
-
19
+
=
28
2+8
=
10
1+0
1
6
I
M
A
G
E
S
-
-
-
-
-
--
-
-
-
-
4
1
7
5
-
+
=
17
1+7
=
8
=
8
-
-
13
1
7
5
-
+
=
26
2+6
=
8
=
8
6
I
M
A
G
E
S
-
-
-
-
-
--
-
-
-
9
13
1
7
5
19
+
=
54
5+4
=
9
=
9
-
9
4
1
7
5
1
+
=
27
2+7
=
9
=
9
6
I
M
A
G
E
S
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
-
-
4
-
-
-
-
=
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
-
-
-
-
7
-
-
=
-
7
occurs
x
1
=
7
-
9
-
-
-
-
-
=
-
9
occurs
x
1
=
9
6
I
M
A
G
E
S
-
-
26
-
-
6
-
27
-
9
-
-
-
-
-
-
-
2+6
-
-
-
-
2+7
6
I
M
A
G
E
S
-
-
8
-
-
6
-
9
-
9
4
1
7
5
1
-
-
-
-
-
--
-
-
6
I
M
A
G
E
S
-
-
8
-
-
6
-
9

 

 

-
7
P
L
A
S
M
A
S
-
-
-
-
-
--
-
-
-
-
-
-
-
1
-
-
1
+
=
2
-
=
2
=
2
-
-
-
-
-
19
-
-
19
+
=
38
3+8
=
11
1+1
2
-
7
P
L
A
S
M
A
S
-
-
-
-
-
--
-
-
-
-
7
3
1
-
4
1
-
+
=
16
1+6
=
7
=
7
-
-
16
12
1
-
13
1
-
+
=
43
4+3
=
7
=
7
-
7
P
L
A
S
M
A
S
-
-
-
-
-
--
-
-
-
-
16
12
1
19
13
1
19
+
=
81
8+1
=
9
=
9
-
-
7
3
1
1
4
1
1
+
=
18
1+8
=
9
=
9
-
7
P
L
A
S
M
A
S
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
1
1
-
-
1
occurs
x
4
=
4
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
--
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
--
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
30
7
P
L
A
S
M
A
S
-
-
15
-
-
7
-
18
3+0
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
-
1+8
3
7
P
L
A
S
M
A
S
-
-
6
-
-
7
-
9

 

 

7
P
L
A
S
M
A
S
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
1
+
=
2
-
=
2
=
2
-
-
-
-
19
-
-
19
+
=
38
3+8
=
11
1+1
2
7
P
L
A
S
M
A
S
-
-
-
-
-
--
-
-
-
7
3
1
-
4
1
-
+
=
16
1+6
=
7
=
7
-
16
12
1
-
13
1
-
+
=
43
4+3
=
7
=
7
7
P
L
A
S
M
A
S
-
-
-
-
-
--
-
-
-
16
12
1
19
13
1
19
+
=
81
8+1
=
9
=
9
-
7
3
1
1
4
1
1
+
=
18
1+8
=
9
=
9
7
P
L
A
S
M
A
S
-
-
-
-
-
-
-
-
-
-
-
1
1
-
1
1
-
-
1
occurs
x
4
=
4
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
7
P
L
A
S
M
A
S
-
-
15
-
-
7
-
18
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
-
1+8
7
P
L
A
S
M
A
S
-
-
6
-
-
7
-
9

 

 

7
SPHERES
90
36
9
5
ROUND
72
27
9
7
BUBBLES
63
18
9

 

 

4
BALL
27
9
9
5
BALLS
46
10
1

 

 

4
BALL
--
-
-
-
B+A
3
3
3
-
L
12
3
3
-
L
12
3
3
4
BALL
27
9
9
-
-
2+7
-
-
4
BALL
9
9
9

 

 

5
ROUND
--
-
-
-
R
18
9
9
-
O+U
36
9
9
-
N+D
18
9
9
5
ROUND
72
27
27
-
-
7+2
2+7
2+7
5
ROUND
9
9
9

 

 

7
SPHERES
--
-
-
-
S+P+H+E
48
30
3
-
R
18
9
9
-
E+S
24
15
6
7
SPHERES
90
54
18
-
-
9+0
5+4
1+8
7
SPHERES
9
9
9

 

 

7
BUBBLES
--
-
-
-
B+U+B+B
27
9
9
-
L+E+S
36
18
9
7
BUBBLES
63
27
18
-
-
6+3
2+7
1+8
7
BUBBLES
9
9
9

 

 

-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
8
-
-
6
+
=
14
1+4
=
5
=
5
-
-
26
-
-
15
+
=
41
4+1
=
5
=
5
-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
-
5
9
-
+
=
14
1+4
=
5
=
5
-
-
-
5
18
-
+
=
23
2+3
=
5
=
5
-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
26
5
18
15
+
=
64
6+4
=
10
1+0
1
-
-
8
5
9
6
+
=
28
2+8
=
10
1+0
1
-
4
Z
E
R
O
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
-
3
-
-
-
-
-
4
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
17
4
Z
E
R
O
-
-
28
-
-
4
-
28
1+7
-
-
-
9
-
-
-
2+8
-
-
-
-
2+8
8
4
Z
E
R
O
-
-
10
-
-
4
-
10
-
-
-
-
-
-
-
-
1+0
-
-
-
-
1+0
8
4
Z
E
R
O
-
-
1
-
-
4
-
1

 

 

-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
8
-
-
6
+
=
14
1+4
=
5
=
5
-
-
26
-
-
15
+
=
41
4+1
=
5
=
5
-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
-
5
9
-
+
=
14
1+4
=
5
=
5
-
-
-
5
18
-
+
=
23
2+3
=
5
=
5
-
4
Z
E
R
O
-
-
-
-
-
--
-
-
-
-
26
5
18
15
+
=
64
6+4
=
10
1+0
1
-
-
8
5
9
6
+
=
28
2+8
=
10
1+0
1
-
4
Z
E
R
O
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
-
3
-
-
-
-
-
4
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
17
4
Z
E
R
O
-
-
28
-
-
4
-
28
1+7
-
-
-
9
-
-
-
2+8
-
-
-
-
2+8
8
4
Z
E
R
O
-
-
10
-
-
4
-
10
-
-
-
-
-
-
-
-
1+0
-
-
-
-
1+0
8
4
Z
E
R
O
-
-
1
-
-
4
-
1

 

 

MIN DOTH DREAM WHAT DOTH MIN MEAN

 

-
MIND
-
-
-
3
M+I+N
36
18
9
1
D
4
4
4
4
MIND
40
22
13
-
-
4+0
2+2
1+3
4
MIND
4
4
4

 

 

`-
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
6
5
-
5
-
1
1
+
=
18
1+8
=
9
=
9
-
9
-
-
15
14
-
14
-
19
19
+
=
81
8+1
=
9
=
9
-
9
-
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
-
5
-
-
+
=
10
1+0
=
1
=
1
-
1
-
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
15
14
5
14
5
19
19
+
=
91
9+1
=
10
1+0
1
-
1
-
-
6
5
5
5
5
1
1
+
=
28
2+8
=
10
1+0
1
-
1
-
7
O
N
E
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
5
5
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
33
7
O
N
E
N
E
S
S
-
-
12
-
-
7
-
28
-
10
3+3
-
-
-
-
-
-
-
-
-
-
1+2
-
-
-
-
2+8
-
1+0
6
7
O
N
E
N
E
S
S
-
-
3
-
-
7
-
10
-
1
-
-
6
5
5
5
5
1
1
-
-
-
-
-
--
-
1+0
-
-
6
7
O
N
E
N
E
S
S
-
-
3
-
-
7
-
1
-
1

 

 

7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
6
5
-
5
-
1
1
+
=
18
1+8
=
9
=
9
-
9
-
15
14
-
14
-
19
19
+
=
81
8+1
=
9
=
9
-
9
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
5
-
-
+
=
10
1+0
=
1
=
1
-
1
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
15
14
5
14
5
19
19
+
=
91
9+1
=
10
1+0
1
-
1
-
6
5
5
5
5
1
1
+
=
28
2+8
=
10
1+0
1
-
1
7
O
N
E
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
=
2
-
-
5
5
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
O
N
E
N
E
S
S
-
-
12
-
-
7
-
28
-
10
-
-
-
-
-
-
-
-
-
-
1+2
-
-
-
-
2+8
-
1+0
7
O
N
E
N
E
S
S
-
-
3
-
-
7
-
10
-
1
-
6
5
5
5
5
1
1
-
-
-
-
-
--
-
1+0
-
-
7
O
N
E
N
E
S
S
-
-
3
-
-
7
-
1
-
1

 

 

-
15
T
H
E
-
O
N
E
N
E
S
S
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
6
5
-
5
-
1
1
-
6
-
-
-
6
-
+
=
38
3+8
=
11
1+1
2
-
2
-
-
-
8
-
-
15
14
-
14
-
19
19
-
15
-
-
-
15
-
+
=
119
1+1+9
=
11
1+1
2
-
2
-
15
T
H
E
-
O
N
E
N
E
S
S
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
-
-
5
-
5
-
-
-
-
6
-
7
-
4
+
=
34
3+4
=
7
=
7
-
7
-
-
20
-
5
-
-
-
5
-
5
-
-
-
-
6
-
7
-
4
+
=
52
5+2
=
7
=
7
-
7
-
15
T
H
E
-
O
N
E
N
E
S
S
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
15
14
5
14
5
19
19
-
15
6
-
7
15
4
+
=
171
1+7+1
=
9
=
9
-
9
-
-
2
8
5
-
6
5
5
5
5
1
1
-
6
6
-
7
6
4
+
=
72
7+2
=
9
=
9
-
9
-
15
T
H
E
-
O
N
E
N
E
S
S
-
O
F
-
G
O
D
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
-
-
-
-
-
-
-
1
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x
2
=
2
=
2
-
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2
-
-
-
-
-
-
-
-
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x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
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4
-
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4
occurs
x
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4
-
-
-
-
5
-
-
5
5
5
5
-
-
-
-
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-
-
-
5
occurs
x
5
=
25
2+5
7
-
-
-
-
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-
6
-
-
-
-
-
-
-
6
6
-
-
6
-
-
-
6
occurs
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=
24
2+4
6
-
-
-
-
-
-
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-
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-
-
-
-
-
-
-
7
-
-
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7
-
-
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8
-
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-
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-
-
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-
-
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8
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1
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8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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15
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-
-
33
-
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15
-
72
-
36
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-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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7+2
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9
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2
8
5
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6
5
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5
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6
6
-
7
6
4
4
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9

 

 

.NEMATODE

 

What are Nematodes? | Department of Nematology

https://nematology.ucr.edu › about › what-are-nematodes

Why are nematodes important? ... Most nematodes feed on bacteria, fungi, or other microscopic creatures. As such, they are a major component of soil and sediment ...
Where are all these worms?

Nematodes have been reported from every continent on earth and occur in deserts, swamps, the oceans, the tropics and Antarctica. Usually nematodes are invisible to all but a few specialists because most are microscopic and transparent.

How many nematodes are there?

Although estimated numbers of species are in the millions, only a few thousand have been named; almost any shovel full of soil, freshwater or marine sediment is likely to have thousands of worms including new species.

Why are nematodes important?

Most nematodes feed on bacteria, fungi, or other microscopic creatures. As such, they are a major component of soil and sediment ecosystems. One species that feeds on soil bacteria, Caenorhabditis elegans, has gained fame as a research model: three specialists on the biology of this worm are the 2002 recipients of the Nobel prize in medicine.

A small fraction of all nematode species are parasites of humans, livestock or agricultural crops. Consequently, these have attracted the most attention from Nematology researchers. For example, root-knot, cyst and lesion nematodes are pests of a wide variety of crops and are annually responsible for billions of dollars of crop losses. Well known animal parasites with health and economic impact include pinworms, hookworms, trichina and dog heartworm.

Certain parasitic nematodes are helpful, including those that attack insects and are used to manage some harmful insects.

The name of the group Nematoda, informally called "nematodes", came from Nematoidea, originally defined by Karl Rudolphi (1808),[30] from Ancient Greek ??µa (nêma, nêmatos, 'thread') and -eid?? (-eides, 'species'). It was treated as family Nematodes by Burmeister (1837).[30]

The nematodes or roundworms constitute the phylum Nematoda (also called Nemathelminthes), with plant-parasitic nematodes being known as eelworms.

Nematodes are microscopic creatures, that act as parasites on other insects. They release bacteria into the host's body to kill them. ... Unlike a chemical spray, which may drift off target, nematodes are specific to the host pest, so that other wildlife is not affected. You can buy nematodes online.

 

NEMATODE

Nematode - Wikipedia
https://en.wikipedia.org › wiki › Nematode

The nematodes or roundworms constitute the phylum Nematoda (also called Nemathelminthes), with plant-parasitic nematodes being known as eelworms.

?Caenorhabditis elegans · ?Hox gene · ?Nematomorpha · ?AscarididaeNematode species can be difficult to distinguish from one another. Consequently, estimates of the number of nematode species described to date vary by author and may change rapidly over time. A 2013 survey of animal biodiversity published in the mega journal Zootaxa puts this figure at over 25,000.[6][7] Estimates of the total number of extant species are subject to even greater variation. A widely referenced[8] article published in 1993 estimated there may be over 1 million species of nematode.[9] A subsequent publication vigorously challenged this claim on the grounds that it is unsupported by fact, estimating the figure to be as low as 40,000 species.[10] Although the highest estimates (up to 100 million species) have since been deprecated, estimates supported by rarefaction curves,[11][12] together with the use of DNA barcoding[13] and the increasing acknowledgment of widespread cryptic species among nematodes,[14] have placed the figure closer to 1 million species.[15]

Nematodes have successfully adapted to nearly every ecosystem: from marine (salt) to fresh water, soils, from the polar regions to the tropics, as well as the highest to the lowest of elevations (including mountains). They are ubiquitous in freshwater, marine, and terrestrial environments, where they often outnumber other animals in both individual and species counts, and are found in locations as diverse as mountains, deserts, and oceanic trenches. They are found in every part of the earth's lithosphere,[16] even at great depths, 0.9–3.6 km (3,000–12,000 ft) below the surface of the Earth in gold mines in South Africa.[17][18][19][20][21] They represent 90% of all animals on the ocean floor.[22] In total, 4.4 × 1020 nematodes inhabit the Earth's topsoil, or approximately 60 billion for each human, with the highest densities observed in tundra and boreal forests.[23] Their numerical dominance, often exceeding a million individuals per square meter and accounting for about 80% of all individual animals on earth, their diversity of lifecycles, and their presence at various trophic levels point to an important role in many ecosystems.[23][24] They have been shown to play crucial roles in polar ecosystems.[25][26] The roughly 2,271 genera are placed in 256 families.[27] The many parasitic forms include pathogens in most plants and animals. A third of the genera occur as parasites of vertebrates; about 35 nematode species occur in humans.[27]

Nathan Cobb, a nematologist, described the ubiquity of nematodes on Earth as thus:

In short, if all the matter in the universe except the nematodes were swept away, our world would still be dimly recognizable, and if, as disembodied spirits, we could then investigate it, we should find its mountains, hills, vales, rivers, lakes, and oceans represented by a film of nematodes. The location of towns would be decipherable since, for every massing of human beings, there would be a corresponding massing of certain nematodes. Trees would still stand in ghostly rows representing our streets and highways. The location of the various plants and animals would still be decipherable, and, had we sufficient knowledge, in many cases even their species could be determined by an examination of their erstwhile nematode parasites.[28]

The term is from Greek ??µat?d?? (plural ??µat?de??), ntr. ??µat?de? (plural ??µat?d?);[29] Latin: Nematoda.
Etymology?[edit]

The word nematode comes from the Modern Latin compound of nemat- "thread" (from Greek nema, genitive nematos "thread," from stem of nein "to spin"; see needle) + -odes "like, of the nature of" (see -oid).
History?[edit]

In 1758, Linnaeus described some nematode genera (e.g., Ascaris), then included in the Vermes.

The name of the group Nematoda, informally called "nematodes", came from Nematoidea, originally defined by Karl Rudolphi (1808),[30] from Ancient Greek ??µa (nêma, nêmatos, 'thread') and -eid?? (-eides, 'species'). It was treated as family Nematodes by Burmeister (1837).[30]

At its origin, the "Nematoidea" erroneously included Nematodes and Nematomorpha, attributed by von Siebold (1843). Along with Acanthocephala, Trematoda, and Cestoidea, it formed the obsolete group Entozoa,[31] created by Rudolphi (1808).[32] They were also classed along with Acanthocephala in the obsolete phylum Nemathelminthes by Gegenbaur (1859).

In 1861, K. M. Diesing treated the group as order Nematoda.[30] In 1877, the taxon Nematoidea, including the family Gordiidae (horsehair worms), was promoted to the rank of phylum by Ray Lankester. The first clear distinction between the nemas and gordiids was realized by Vejdovsky when he named a group to contain the horsehair worms the order Nematomorpha. In 1919, Nathan Cobb proposed that nematodes should be recognized alone as a phylum.[33] He argued they should be called "nema" in English rather than "nematodes" and defined the taxon Nemates (later emended as Nemata, Latin plural of nema), listing Nematoidea sensu restricto as a synonym.

However, in 1910, Grobben proposed the phylum Aschelminthes and the nematodes were included in as class Nematoda along with class Rotifera, class Gastrotricha, class Kinorhyncha, class Priapulida, and class Nematomorpha (The phylum was later revived and modified by Libbie Henrietta Hyman in 1951 as Pseudoceolomata, but remained similar). In 1932, Potts elevated the class Nematoda to the level of phylum, leaving the name the same. Despite Potts' classification being equivalent to Cobbs', both names have been used (and are still used today) and Nematode became a popular term in zoological science.[34]

Since Cobb was the first to include nematodes in a particular phylum separated from Nematomorpha, some researchers consider the valid taxon name to be Nemates or Nemata, rather than Nematoda,[35] because of the zoological rule that gives priority to the first used term in case of synonyms.

Phylogeny?[edit]

The phylogenetic relationships of the nematodes and their close relatives among the protostomian Metazoa are unresolved. Traditionally, they were held to be a lineage of their own, but in the 1990s, they were proposed to form the group Ecdysozoa together with moulting animals, such as arthropods. The identity of the closest living relatives of the Nematoda has always been considered to be well resolved. Morphological characters and molecular phylogenies agree with placement of the roundworms as a sister taxon to the parasitic Nematomorpha; together, they make up the Nematoida. Along with the Scalidophora (formerly Cephalorhyncha), the Nematoida form the clade Cycloneuralia, but much disagreement occurs both between and among the available morphological and molecular data. The Cycloneuralia or the Introverta—depending on the validity of the former—are often ranked as a superphylum.[36]

Nematode systematics?[edit]

Due to the lack of knowledge regarding many nematodes, their systematics is contentious. An early and influential classification was proposed by Chitwood and Chitwood[37]—later revised by Chitwood[38]—who divided the phylum into two classes—Aphasmidia and Phasmidia. These were later renamed Adenophorea (gland bearers) and Secernentea (secretors), respectively.[39] The Secernentea share several characteristics, including the presence of phasmids, a pair of sensory organs located in the lateral posterior region, and this was used as the basis for this division. This scheme was adhered to in many later classifications, though the Adenophorea were not in a uniform group.

Initial studies of incomplete DNA sequences[40] suggested the existence of five clades:[41]
Dorylaimida
Enoplia
Spirurina
Tylenchina
Rhabditina

The Secernentea seem to be a natural group of close relatives, while the "Adenophorea" appear to be a paraphyletic assemblage of roundworms that retain a good number of ancestral traits. The old Enoplia do not seem to be monophyletic, either, but do contain two distinct lineages. The old group "Chromadoria" seems to be another paraphyletic assemblage, with the Monhysterida representing a very ancient minor group of nematodes. Among the Secernentea, the Diplogasteria may need to be united with the Rhabditia, while the Tylenchia might be paraphyletic with the Rhabditia.[42]

The understanding of roundworm systematics and phylogeny as of 2002 is summarised below:

Phylum Nematoda
Basal order Monhysterida
Class Dorylaimida
Class Enoplea
Class Secernentea Subclass Diplogasteria (disputed)
Subclass Rhabditia (paraphyletic?)
Subclass Spiruria
Subclass Tylenchia (disputed)

"Chromadorea" assemblage

Later work has suggested the presence of 12 clades.[43] The Secernentea—a group that includes virtually all major animal and plant 'nematode' parasites—apparently arose from within the Adenophorea.

In 2019, a study identified one conserved signature indel (CSI) found exclusively in members of the phylum Nematoda through comparative genetic analyses.[44] The CSI consists of a single amino acid insertion within a conserved region of a Na(+)/H(+) exchange regulatory factor protein NRFL-1 and is a molecular marker that distinguishes the phylum from other species.[44]

A major effort by a collaborative wiki called 959 Nematode Genomes is underway to improve the systematics of this phylum.[45]

A complete checklist of the world's nematode species can be found in the World Species Index: Nematoda.[46]

An analysis of the mitochondrial DNA suggests that the following groupings are valid[47]
subclass Dorylaimia
orders Rhabditida, Trichinellida and Mermithida
suborder Rhabditina
infraorders Spiruromorpha and Oxyuridomorpha

Anatomy?[edit]

Internal anatomy of a male C. elegans nematode
Nematodes are very small, slender worms: typically about 5 to 100 µm thick, and 0.1 to 2.5 mm long.[48] The smallest nematodes are microscopic, while free-living species can reach as much as 5 cm (2 in), and some parasitic species are larger still, reaching over 1 m (3 ft) in length.[49]:271 The body is often ornamented with ridges, rings, bristles, or other distinctive structures.[50]

The head of a nematode is relatively distinct. Whereas the rest of the body is bilaterally symmetrical, the head is radially symmetrical, with sensory bristles and, in many cases, solid 'head-shields' radiating outwards around the mouth. The mouth has either three or six lips, which often bear a series of teeth on their inner edges. An adhesive 'caudal gland' is often found at the tip of the tail.[51]

The epidermis is either a syncytium or a single layer of cells, and is covered by a thick collagenous cuticle. The cuticle is often of a complex structure and may have two or three distinct layers. Underneath the epidermis lies a layer of longitudinal muscle cells. The relatively rigid cuticle works with the muscles to create a hydroskeleton, as nematodes lack circumferential muscles. Projections run from the inner surface of muscle cells towards the nerve cords; this is a unique arrangement in the animal kingdom, in which nerve cells normally extend fibers into the muscles rather than vice versa.[51]

Digestive system?[edit]

The oral cavity is lined with cuticle, which is often strengthened with structures, such as ridges, especially in carnivorous species, which may bear a number of teeth. The mouth often includes a sharp stylet, which the animal can thrust into its prey. In some species, the stylet is hollow and can be used to suck liquids from plants or animals.[51]

The oral cavity opens into a muscular, sucking pharynx, also lined with cuticle. Digestive glands are found in this region of the gut, producing enzymes that start to break down the food. In stylet-bearing species, these may even be injected into the prey.[51]

No stomach is present, with the pharynx connecting directly to a muscleless intestine that forms the main length of the gut. This produces further enzymes, and also absorbs nutrients through its single-cell-thick lining. The last portion of the intestine is lined by cuticle, forming a rectum, which expels waste through the anus just below and in front of the tip of the tail. The movement of food through the digestive system is the result of the body movements of the worm. The intestine has valves or sphincters at either end to help control the movement of food through the body.[51]

Excretory system?[edit]

Nitrogenous waste is excreted in the form of ammonia through the body wall, and is not associated with any specific organs. However, the structures for excreting salt to maintain osmoregulation are typically more complex.[51]

In many marine nematodes, one or two unicellular 'renette glands' excrete salt through a pore on the underside of the animal, close to the pharynx. In most other nematodes, these specialized cells have been replaced by an organ consisting of two parallel ducts connected by a single transverse duct. This transverse duct opens into a common canal that runs to the excretory pore.[51]

Nervous system?[edit]

See also: Muscle arms

Four peripheral nerves run along the length of the body on the dorsal, ventral, and lateral surfaces. Each nerve lies within a cord of connective tissue lying beneath the cuticle and between the muscle cells. The ventral nerve is the largest, and has a double structure forward of the excretory pore. The dorsal nerve is responsible for motor control, while the lateral nerves are sensory, and the ventral combines both functions.[51]

The nervous system is also the only place in the nematode body that contains cilia, which are all nonmotile and with a sensory function.[52][53]

At the anterior end of the animal, the nerves branch from a dense, circular nerve (nerve ring) round surrounding the pharynx, and serving as the brain. Smaller nerves run forward from the ring to supply the sensory organs of the head.[51]

The bodies of nematodes are covered in numerous sensory bristles and papillae that together provide a sense of touch. Behind the sensory bristles on the head lie two small pits, or 'amphids'. These are well supplied with nerve cells and are probably chemoreception organs. A few aquatic nematodes possess what appear to be pigmented eye-spots, but whether or not these are actually sensory in nature is unclear.[51]

Reproduction?[edit]

Extremity of a male nematode showing the spicule, used for copulation, bar = 100 µm[54]
Most nematode species are dioecious, with separate male and female individuals, though some, such as Caenorhabditis elegans, are androdioecious, consisting of hermaphrodites and rare males. Both sexes possess one or two tubular gonads. In males, the sperm are produced at the end of the gonad and migrate along its length as they mature. The testis opens into a relatively wide seminal vesicle and then during intercourse into a glandular and muscular ejaculatory duct associated with the vas deferens and cloaca. In females, the ovaries each open into an oviduct (in hermaphrodites, the eggs enter a spermatheca first) and then a glandular uterus. The uteri both open into a common vulva/vagina, usually located in the middle of the morphologically ventral surface.[51]

Reproduction is usually sexual, though hermaphrodites are capable of self-fertilization. Males are usually smaller than females or hermaphrodites (often much smaller) and often have a characteristically bent or fan-shaped tail. During copulation, one or more chitinized spicules move out of the cloaca and are inserted into the genital pore of the female. Amoeboid sperm crawl along the spicule into the female worm. Nematode sperm is thought to be the only eukaryotic cell without the globular protein G-actin.

Eggs may be embryonated or unembryonated when passed by the female, meaning their fertilized eggs may not yet be developed. A few species are known to be ovoviviparous. The eggs are protected by an outer shell, secreted by the uterus. In free-living roundworms, the eggs hatch into larvae, which appear essentially identical to the adults, except for an underdeveloped reproductive system; in parasitic roundworms, the lifecycle is often much more complicated.[51]

Nematodes as a whole possess a wide range of modes of reproduction.[55] Some nematodes, such as Heterorhabditis spp., undergo a process called endotokia matricida: intrauterine birth causing maternal death.[56] Some nematodes are hermaphroditic, and keep their self-fertilized eggs inside the uterus until they hatch. The juvenile nematodes then ingest the parent nematode. This process is significantly promoted in environments with a low food supply.[56]

The nematode model species C. elegans, C. briggsae, and Pristionchus pacificus, among other species, exhibit androdioecy,[57] which is otherwise very rare among animals. The single genus Meloidogyne (root-knot nematodes) exhibits a range of reproductive modes, including sexual reproduction, facultative sexuality (in which most, but not all, generations reproduce asexually), and both meiotic and mitotic parthenogenesis.

The genus Mesorhabditis exhibits an unusual form of parthenogenesis, in which sperm-producing males copulate with females, but the sperm do not fuse with the ovum. Contact with the sperm is essential for the ovum to begin dividing, but because no fusion of the cells occurs, the male contributes no genetic material to the offspring, which are essentially clones of the female.[51]

Free-living species?[edit]

Different free-living species feed on materials as varied as bacteria, algae, fungi, small animals, fecal matter, dead organisms, and living tissues. Free-living marine nematodes are important and abundant members of the meiobenthos. They play an important role in the decomposition process, aid in recycling of nutrients in marine environments, and are sensitive to changes in the environment caused by pollution. One roundworm of note, C. elegans, lives in the soil and has found much use as a model organism. C. elegans has had its entire genome sequenced, the developmental fate of every cell determined, and every neuron mapped.

Parasitic species?[edit]

Eggs (mostly nematodes) from stools of wild primates
Nematodes that commonly parasitise humans include ascarids (Ascaris), filarias, hookworms, pinworms (Enterobius), and whipworms (Trichuris trichiura). The species Trichinella spiralis, commonly known as the 'trichina worm', occurs in rats, pigs, bears, and humans, and is responsible for the disease trichinosis. Baylisascaris usually infests wild animals, but can be deadly to humans, as well. Dirofilaria immitis is known for causing heartworm disease by inhabiting the hearts, arteries, and lungs of dogs and some cats. Haemonchus contortus is one of the most abundant infectious agents in sheep around the world, causing great economic damage to sheep. In contrast, entomopathogenic nematodes parasitize insects and are mostly considered beneficial by humans, but some attack beneficial insects.

One form of nematode is entirely dependent upon fig wasps, which are the sole source of fig fertilization. They prey upon the wasps, riding them from the ripe fig of the wasp's birth to the fig flower of its death, where they kill the wasp, and their offspring await the birth of the next generation of wasps as the fig ripens.

Colorized electron micrograph of soybean cyst nematode (Heterodera sp.) and egg
A newly discovered parasitic tetradonematid nematode, Myrmeconema neotropicum, apparently induces fruit mimicry in the tropical ant Cephalotes atratus. Infected ants develop bright red gasters (abdomens), tend to be more sluggish, and walk with their gasters in a conspicuous elevated position. These changes likely cause frugivorous birds to confuse the infected ants for berries, and eat them. Parasite eggs passed in the bird's feces are subsequently collected by foraging C. atratus and are fed to their larvae, thus completing the lifecycle of M. neotropicum.[58]

Similarly, multiple varieties of nematodes have been found in the abdominal cavities of the primitively social sweat bee, Lasioglossum zephyrus. Inside the female body, the nematode hinders ovarian development and renders the bee less active, thus less effective in pollen collection.[59]

Plant-parasitic nematodes include several groups causing severe crop losses. The most common genera are Aphelenchoides (foliar nematodes), Ditylenchus, Globodera (potato cyst nematodes), Heterodera (soybean cyst nematodes), Longidorus, Meloidogyne (root-knot nematodes), Nacobbus, Pratylenchus (lesion nematodes), Trichodorus, and Xiphinema (dagger nematodes). Several phytoparasitic nematode species cause histological damages to roots, including the formation of visible galls (e.g. by root-knot nematodes), which are useful characters for their diagnostic in the field. Some nematode species transmit plant viruses through their feeding activity on roots. One of them is Xiphinema index, vector of grapevine fanleaf virus, an important disease of grapes, another one is Xiphinema diversicaudatum, vector of arabis mosaic virus.

Other nematodes attack bark and forest trees. The most important representative of this group is Bursaphelenchus xylophilus, the pine wood nematode, present in Asia and America and recently discovered in Europe.

Agriculture and horticulture?[edit]

Depending on its species, a nematode may be beneficial or detrimental to plant health. From agricultural and horticulture perspectives, the two categories of nematodes are the predatory ones, which kill garden pests such as cutworms and corn earworm moths, and the pest nematodes, such as the root-knot nematode, which attack plants, and those that act as vectors spreading plant viruses between crop plants.[60] Plant-parasitic nematodes are often known as eelworms and attack leaves and buds. Predatory nematodes can be bred by soaking a specific recipe of leaves and other detritus in water, in a dark, cool place, and can even be purchased as an organic form of pest control.[citation needed]

Rotations of plants with nematode-resistant species or varieties is one means of managing parasitic nematode infestations. For example, marigolds, grown over one or more seasons (the effect is cumulative), can be used to control nematodes.[61] Another is treatment with natural antagonists such as the fungus Gliocladium roseum. Chitosan, a natural biocontrol, elicits plant defense responses to destroy parasitic cyst nematodes on roots of soybean, corn, sugar beet, potato, and tomato crops without harming beneficial nematodes in the soil.[62] Soil steaming is an efficient method to kill nematodes before planting a crop, but indiscriminately eliminates both harmful and beneficial soil fauna.

The golden nematode Globodera rostochiensis is a particularly harmful variety of nematode pest that has resulted in quarantines and crop failures worldwide. CSIRO has found a 13- to 14-fold reduction of nematode population densities in plots having Indian mustard Brassica juncea green manure or seed meal in the soil.[63]

Epidemiology?[edit]

Disability-adjusted life year for intestinal nematode infections per 100,000 in 2002.

File:Anthelmintic effect of papain on Heligmosomoides bakeri.ogv
Play media

Anthelmintic effect of papain on Heligmosomoides bakeri
A number of intestinal nematodes cause diseases affecting human beings, including ascariasis, trichuriasis, and hookworm disease. Filarial nematodes cause filariases.

Soil ecosystems?[edit]

Further information: Soil ecology

About 90% of nematodes reside in the top 15 cm (6") of soil. Nematodes do not decompose organic matter, but, instead, are parasitic and free-living organisms that feed on living material. Nematodes can effectively regulate bacterial population and community composition—they may eat up to 5,000 bacteria per minute. Also, nematodes can play an important role in the nitrogen cycle by way of nitrogen mineralization.[48]

One group of carnivorous fungi, the nematophagous fungi, are predators of soil nematodes.[64] They set enticements for the nematodes in the form of lassos or adhesive structures.[65][66][67]

Society and culture?[edit]

Nematode worms (C. elegans), part of an ongoing research project conducted on the 2003 Space Shuttle Columbia mission STS-107, survived the re-entry breakup. It is believed to be the first known life form to survive a virtually unprotected atmospheric descent to Earth's surface.[68][69]

See also?[edit]
Biological pest control – Controlling pests using other organisms
Capillaria – Genus of roundworms
List of organic gardening and farming topics
List of parasites of humans – Wikipedia list article
Toxocariasis – Illness of humans caused by larvae of the dog, the cat or the fox roundworm: A helminth infection of humans caused by the dog or cat roundworm, Toxocara canis or Toxocara cati
Worm bagging – Process where nematode eggs hatch within the parent and the larvae proceed to consume and emerge from the parent

 

SECERNENTEA

Due to the lack of knowledge regarding many nematodes, their systematics is contentious. An early and influential classification was proposed by Chitwood and Chitwood[37]—later revised by Chitwood[38]—who divided the phylum into two classes—Aphasmidia and Phasmidia. These were later renamed Adenophorea (gland bearers) and Secernentea (secretors), respectively.[39] The Secernentea share several characteristics, including the presence of phasmids, a pair of sensory organs located in the lateral posterior region, and this was used as the basis for this division.

Secernentea
Animal
Description
Secernentea was a class of nematodes in the Classical Phylogeny System and is no longer in use. This morphological-based classification system has been replaced by the Modern Phylogeny system, where taxonomy assignment is based on small subunit ribosomal DNA. Wikipedia
Higher classification: Roundworms

Scientific name: Secernentea

SECERNENTEA

 

 
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