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THE KEYS OF THE
TEMPLE
David Furlong
1997
Page 93
6
SACRED GEOMETRY AND
THE GIZA PYRAMIDS
"The alignment of the
pyramids Grand Gallery pointed to the centre of one of the
circles
Above the entrance to
Plato's Academy at Athens was the legend 'Let none ignorant
of geometry enter here'. To the ancient Greeks, pure
geometry lay at the heart of all things. It was the way of
reconciling the world of the divine with the form of the
world we see. The golden mean proportion, for example, can
be depicted in terms of geometry but not number. It can be
drawn, but the number that represents it cannot be written
down as it runs to an infinite number of decimal places.
Geometry can be seen as a way of defining what is otherwise
indefinable.
Our knowledge of the use
of pure geometry in ancient Egypt is more tenuous. We do not
have any papyri which give the geometrical equivalent of the
equations of Plato, Thales and Euclid, epitomising ancient
Greek thought. However, Plato considered that Egypt
possessed a profound canon of knowledge based on harmony and
proportion. We can infer that the ancient Egyptians were as
adept with the compass and the rule as their Greek
counterparts. This knowledge would have influenced their art
and architecture. Unravelling how the Egyptians might have
selected the proportions they used is a way of reaching back
into the roots of their civilisation.
/ Page 94 /
The patterns on the
Marlborough Downs also appeared to be based on pure
geometry. Our next step, therefore, must be to draw together
these disparate parts of the puzzle and discover the
underlying geometry that unites them.
SACRED
GEOMETRY
" The term 'sacred
geometry' can be misleading as the fundamentals of
geo-m~tric proportion are found extensively throughout the
natural world as well as in art and architecture. Why are
some elements sacred and not others? There is no easy answer
to this question. Nevertheless, a tradition has grown up
which has placed a special emphasis on certain geometric
relationships and proportions found most commonly in the
design of buildings used for religious purposes. To the
general observer, these pro- portions are simply pleasing.
Artistically, this is analogous to music. Using different
groupings of notes and internals, harmonious or discordant
sounds can be created. Some music, such as Gregorian chants,
can put us more in touch with spiritual feelings. Other
music puts us directly in touch with our emotions. Indeed,
one of the great philosophers, Pythagoras, demonstrated the
links between music, sound, number and form.
Three basic geometric
shapes are central to religious tradition: the circle, the
triangle and the square (Fig. 29). (Diagram omitted)
These were taken to represent three levels of our being:
spirit, mind and body. Like systems of counting, no one
knows who first used a compass. It was probably just a piece
of /
SPIRIT . . . MIND
. . . MATTER or FORM
Fig. 29. The three
basic shapes in geometry
Page
95 /
string
and two pegs,"6ut this development paved the way for a
symbolic exploration into the realm of ideas and forms. All
regular geometric shapes can be derived using a compass.
God, who has sometimes been called the 'Great Geometer', is
often depicted wielding a pair.
'Linked to geometry was
the study of numbers. Whole numbers were considered the
ideal. A completeness was perceived in them; while
frac-tions represented numbers in the stage of becoming. In
this sense, they were sometimes considered as the dynamic
power of the divine moving through creation. Whole numbers
were knowable, but ratios such as pi..." "... could only be
approximations and therefore unknowable. This was the
unfathomable hand of God that permeates all
things.
But while individual
numbers are either rational (whole numbers) or irrational
(fractional numbers), geometry can bridge this divide. The
circle can both represent a rational whole number principle
in its diame-ter as well as an irrational function along its
circumference. A square and its diagonal also produce a
similar phenomenon..."
Page
98
Fig. 34. Pentagon
derived from vesica
(Diagram
omitted)
"...The triangle was seen
as a transitional form between the square and the circle.
Eventually it came to represent a triad of gods and
goddesses, usu- ally, as in Egypt, a father, mother and son.
This concept formed a central pivot of many religious belief
systems and manifests itself in Christianity as the Father,
Son and Holy Ghost. The triangle's most perfect form was
considered to be equilateral, where the sides and the angles
are all equal. Another triangle that was widely used was the
one attributed to Pythagoras, although it certainly predated
him by a very long time. Its
/ Page
99
Fig. 35. Construction
of golden mean from 2 x 1 rectangle
(Diagram
omitted)
Fig 36. Construction
of golden mean from a square and a circle
(Diagram
omitted)
/
sides are in. the whole
number ratio 3:4:5. This triangle produces the simplest
version of a right-angled (90°) triangle with sides
that can be expressed as whole numbers. Because of the
simple numeric ratios employed, it was used in surveying as
well as in art and sculpture. The pyramid of Khafre is based
on it.
The circle, triangle,
square and rectangle form the basis of all sacred
architecture. They were, by tradition, related to each other
by specific pro- portions. These proportions attempted to
portray the inherent harmony of the cosmos. One such
proportion was the 'gnomon' which was defined /
Page 100
Fig.
37.
The construction
of a pentagon
(Diagram omitted)
Fig. 38. The golden
mean proportion within a pentagram.
(Diagram
omitted)
Page 101 /
by Aristotle as any
figure which, when added to an original figure, leaves the
resultant figure similar to the original.' In other words,
the ratios between each additional step are always
maintained. An example of this is the 'golden mean'
proportion which can be expressed through the num-bers 1,
1,2,3, 5, 8, 13,23, and so on, where the ratios between any
two adjoining numbers rapidly converge as you move up the
series. The Fibonacci series is the most well known example
of a gnomic ratio, but others exist.
Robert Lawlor, in his
book Sacred Geometry, gives examples of 'gno- monic' spirals
such as the one based on the Fibonacci series, which is
derived from the ratio of 1 :2. These expanding patterns are
sometimes called 'whirling squares', giving rise to spirals
which are frequently found in the natural world (Fig. 39).
(Figure 39 omitted)
Examining 'gnomons' of
different ratios, I made a significant discovery. One of the
'gnomons' based on the ratio 1:3 is directly relevant to the
Giza
/ Page 102 /
pyramids. It so happens
that from this one ratio the basic proportions of Khufu's,
Khafre's and Menkaure's pyramids can all be derived (Fig.
40). Figure 40 omitted ) The development starts by drawing
three abutting squares in line, to cre-ate a 3 by 1
rectangle. A square is then drawn on the longer side at each
stage of its development.
The first square creates
a rectangle with a ratio of 3:4. Doubling it reveals the
ratio of Khafre's pyramid, 6:4. Adding two more squares, in
turn, to the 3:4 rectangle gives the ratio of Khufu's
pyramid, 7:11.
/
Page 103 /
Another square"gives the
proportions of the pyramid of Menkaure, 11: 18. This device
of adding squares, starting with a 3 by 1 rectangle,
dramati-cally reveals that the pyramids reflect a natural
mathematical progression in their height to base ratios.
Whether through chance or choice they are linked by a
harmonious geometric series.
What could have been so
significant about a 3: 1 ratio? Perhaps it reflected the
symbolism of the Egyptian trinity of Osiris, Isis and Horus.
We will probably never know for sure, but this pattern gives
a valuable insight into Egyptian methods.
This discovery is also in
accordance with what is known about Egyptian design methods
which always seem to have been derived from squared grid
patterns. There are numerous examples existing in Egyptian
art, showing painters and sculptors first drawing a grid on
the wall that is to be painted or carved so that fixed
proportions could be maintained. The simple numeric ratios
of these grids lie at the heart of all the great artistic
achievements of the Egyptians.
This method was also used
by many of the great Renaissance artists, such as Leonardo
da Vinci. In ancient Egypt, this is exemplified in the Great
Pyramid and establishes a further link with the pattern on
the Marlborough Downs.
THE
PARANORMAL
Stuart Gordon
1992
Page388
"LEIBNITZ, Gottfreid
Wilhelm van (1646-1716) This 'greatest intellectual genius
since Aristotle' (Macneile Dixon) claimed that matter is but
another name for energy, and that space and time are
inseparable. He said, 'The world is not a machine.
Everything in it is force, life, thought, desire. The world,
in brief. . . is a living society'.
Page389
Believing that all living
individuals always have existed and always will exist, he
regarded death as a longer, profounder version of sleep, in
which the continuity of the individual life is not wholly
broken. He held the universe to be composed of monads;
living and active beings who reflect the universe each from
its own angle, each in its own degree.
From his Monadology:
'There is nothing waste, nothing sterile, nothing dead in
the universe; no chaos, no confusions, save in
appearance.'
LEONARDO DA VINCI
(1452-1519) Leonardo, whose genius led him in so many
prophetic directions, was clearly a believer in
reincarnation. In his Notebooks he wrote: 'Behold now the
hope and desire to go back to our own country, and to return
to our former state, how like it is to the moth with the
light! . . . this longing is the quintessence and spirit of
the elements, which, finding itself imprisoned within the
life of the human body, desires continually to return to its
source.' And again: 'Read me, 0 Reader, if you find delight
in me, because very seldom shall I come back into this
world.'
THE KEYS OF THE
TEMPLE
David Furlong
1997
Page
88
"The angle of slope of
Khafre's pyramid is the same as that of the pyra- mid of
Pepi II, one of the kings of the 6th Dynasty who reigned
from 2278 to 2184BC..."
2278 minus 2184 =
94
"The angle of slope of
Khafre's pyramid is the same as that of the pyra- mid of
Pepi II, one of the kings of the 6th Dynasty who reigned
from 2278 to 2184BC. This pyramid is now in ruins, but the
angle of slope has been deduced from what remains of the
outer casings. The construction of the later Egyptian
pyramids was inferior to the Giza group and they have
suffered greatly through the passage of time. Many have now
collapsed into heaps of rubble. However, in Khafre's case,
the angle of his pyramid, (based on the
3:4:5
triangle) is highlighted in three of the problems (num-bers
57,
58
and
59)
posed on pyramids in the Rhind Mathematical Papyrus. So it
was a ratio that was well known to the ancient
Egyptians.
In fairness to those
Egyptologists who maintain that the ancient Egyptians did
not know about the
3:4:5
triangle, the length of the hypotenuse - 5 - is never given.
But mathematical problems, and those involving the pyramids,
are always calculated in terms of the seked of the angle,
the ratio of height to base. In the case of the
3:4:5
triangle, the seked is the ratio of 3 to 4. But as no
mention was ever made of the length of the hypotenuse, it
was inferred that the Egyptians had never worked out the
length of the third side.
Are we really supposed to
believe that a people who could design and build a monument
to the precision shown in the Great Pyramid, or the pyramid
of Khafre, inv,olving exact measurements over considerable
dis- tances, would never have measured or worked out the
lengths of the hypotenuses of the triangles they were using?
Surely, any people seeking precision of measurement would
check all lengths as a matter of course as part of their
general exploration of number, form and geometry. It would
be an essential part of their working methods. So, I would
maintain that simply by using the ratio
3:4
in their building
plans, they implicitly knew the length of the third side.
"
"...However, in Khafre's
case, the angle of his pyramid, (based on the
3:4:5
triangle) is highlighted in three of the problems (num-bers
57,
58
and
59)
posed on pyramids in the Rhind Mathematical Papyrus. So it
was a ratio that was well known to the ancient
Egyptians..."
"In the case of the
3:4:5
triangle, the seked is the ratio of
3
to
4.
THE GROWTH OF
SCIENCE
A.P. Rossiter
1939
Page 15
"The Egyptians,…"
"…made good observations on the stars and were able to
say when the sun or moon would become dark in an eclipse (a
most surprising event even in our times), and when the land
would be covered by the waters of the Nile: they were expert
at building and made some discoveries about the relations of
lines and angles - among them one very old rule for getting
a right-angle by stretching out knotted cords with 5, 4, and
3 units between the knots.
" knotted cords
with
5,
4,
and
3
units between the knots. "
Reight wah
scribe, if there iz a point to labour, set up the labourer,
and labour that point.
The scribe quick to
noblesse oblige, counted the number of single letters in
each of three words.
5 . . . 4 . . .
3
EARTHMOONSUN
3
THREESUN
4
FOURMOON
5
FIVEEARTH
Page 98
This concept formed a
central pivot of many religious belief systems and manifests
itself in Christianity as the
Father,
Son
and
Holy
Ghost.
The triangle's most
perfect form was considered to be equilateral, where the
sides and the angles are all equal. Another triangle that
was widely used was the one attributed to Pythagoras,
although it certainly predated him by a very long time.Its
/
Page
99
sides are in. the whole
number ratio
3:4:5.
This triangle produces
the simplest version of a right-angled
(90°)
triangle with sides that can be expressed as whole
numbers.
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3
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19
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21
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14
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+
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54
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5+4
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=
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9
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1
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3
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5
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+
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9
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NINE
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9
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4
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13
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15
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15
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14
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+
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=
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57
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5+7
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=
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12
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1+2
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=
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3
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4
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6
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6
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5
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21
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2+1
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3
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THREE
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3
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5
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5
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18
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20
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8
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52
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5+2
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7
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5
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1
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9
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2
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8
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+
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=
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25
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2+5
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=
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7
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SEVEN
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7
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THE KEYS OF THE
TEMPLE
David Furlong
1997
Page 94
SACRED
GEOMETRY
"...The triangle was seen
as a transitional form between the square and the circle.
Eventually it came to represent a triad of gods and
goddesses, usu-ally, as in Egypt, a father, mother and son.
This concept formed a central pivot of many religious belief
systems and manifests itself in Christianity as the Father,
Son and Holy Ghost. The triangle's most perfect form was
considered to be equilateral, where the sides and the angles
are all equal. Another triangle that was widely used was the
one attributed to Pythagoras, although it certainly predated
him by a very long time. Its
/ Page
99 /
sides are in. the whole
number ratio
3:4:5.
This triangle produces the simplest version of a
right-angled
(90°)
triangle with sides that can be expressed as whole
numbers.
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THE
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MAGIKAL
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ALPHABETS
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NUMERICAL
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STATUS
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S
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SATURN
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93
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9+3
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12
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=
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1+2
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=
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3
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U
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URANUS
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94
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9+4
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13
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=
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1+3
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=
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4
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N
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NEPTUNE
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95
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9+5
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14
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=
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1+4
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=
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5
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6
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19
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1
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20
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21
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18
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14
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+
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=
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93
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9+3
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12
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1+2
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=
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3
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1+9
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2+0
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2+1
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1+8
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1+4
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10
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2
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3
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9
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5
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1+0
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=
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1
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1
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1
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2
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3
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9
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5
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=
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21
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2+1
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=
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3
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THREE
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3
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6
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19
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1
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20
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21
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18
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14
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+
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=
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93
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9+3
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=
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12
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1+2
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=
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3
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1
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1
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2
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3
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9
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5
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+
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21
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2+1
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3
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THREE
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3
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6
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21
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18
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1
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14
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21
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19
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=
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94
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9+4
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13
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1+3
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4
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2+1
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1+8
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1+4
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2+1
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1+9
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3
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9
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1+0
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1
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1
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3
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9
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1
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5
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3
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1
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22
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2+2
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=
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4
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FOUR
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4
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21
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18
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1
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14
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21
|
19
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+
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=
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94
|
9+4
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=
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13
|
1+3
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=
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4
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3
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9
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1
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5
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3
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1
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+
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=
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22
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2+2
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=
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4
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FOUR
|
4
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