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"... What was this meticulous interest in the angle of 26°, and was it a coincidence that it amounted to half of the angle of inclination of the pyramid's sides - 52°.10

The reader may recall the significance of this angle. It was a key ingredient of the sophisticated and advanced formula by which the design of the Great Pyramid had been made to correspond precisely to the dynamics of spherical geometry. Thus the original height of the monument (481.3949 feet), and the perimeter of its base (3023.16 feet), stood in the same ratio to each other as did the radius of a sphere to its circumference. This ratio was 2pi (2 x 3.14) and to express it the builders had been obliged to specify the tricky and idiosyncratic angle of 52° for the pyramid's sides (since any greater or lesser slope would have meant a different height-to-perimeter ratio).

In Chapter Twenty-three we saw that the so-called Pyramid of the Sun at Teotihuacan in Mexico also expressed a knowledge and deliberate use of the transcendental number pi; in its case the height (233.5 feet) stood in a relationship of 4pi to the perimeter of its base (2932.76 feet).11

The crux, therefore, was that the most remarkable monument of Ancient Egypt and the most remarkable monument of Ancient Mexico both incorporated pi relationships long before and far away from the official 'discovery' of this transcendental number by the Greeks.12 Moreover, the evidence invited the conclusion that something was being signalled by the use of pi - almost certainly the same thing in both cases.

Not for the first time, and not for the last, I was overwhelmed by a sense of contact with an ancient intelligence, not necessarily Egyptian or Mexican, which had found a way to reach out across the ages and draw people towards it like a beacon. Some might look for treasure; others, captivated by the deceptively simple manner in which the builders had used pi to demonstrate their mastery of the secrets of transcendental numbers, might be inspired to search for further mathematical epiphanies.

Bent almost double, my back brushing against the polished limestone ceiling, it was with such thoughts in my mind that I began to scramble up the 26° slope of the ascending. corridor, w:hich seemed to penetrate the vast bulk of the six million ton buildmg like a '

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trigonometrical device. After I had banged my head on-its ceiling a couple of times, however, I began to wonder why the ingenious people who'd designed it hadn't made it two or three feet higher. If they could erect a monument like this in the first place (which they obviously could) and equip it with corridors, surely it would not have been beyond their capabilities to make those corridors roomy enough to stand up in? Once again I was tempted to conclude that it was the result of a deliberate decision by the pyramid builders: they had made the ascending corridor this way because they had wanted it this way (rather than because such a design had been forced upon them.)

Was there motive in the apparent madness of these archaic mind games?

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Reaching the top of the Grand Gallery, I clambered over a chunky granite step about three feet high. I remembered that it lay, like the roof of the Queen's Chamber, exactly along the east-west axis of the Great Pyramid, And therefore marked the point of transition between the northern and southern halves of the monument.' Somewhat like an altar in appearance, the step also provided a solid horizontal platform immediately in front of the low square tunnel that served as the entrance to the King's Chamber.

Pausing for a moment, I looked back down the Gallery, taking in once again its lack of decoration, its lack of religious iconography, and its absolute lack of any of the recognizable symbolism normally associated with the archaic belief system of the Ancient Egyptians. All that registered upon the eye, along the entire 153-foot length of this magnificent geometrical cavity, was its disinterested regularity and its stark machinelike simplicity.

Looking up, I could just make out the opening of a dark aperture, chiselled into the top of the eastern wall above my head. Nobody knew when or by whom this foreboding hole had been cut, or how deep it had originally penetrated. It led to the first of the five relieving chambers above the King's Chamber and had been extended in 1837 when Howard Vyse had used it to break through to the remaining four. Looking down again, I could just make out the point at the bottom of the Gallery's western wall where the near-vertical well-

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shaft began its precipitous 160 foot descent through the core of the pyramid to join the descending corridor far below ground-level.

Why would such a complicated apparatus of pipes and passageways have been required? At first sight it didn't make sense. But then nothing about the Great Pyramid did make much sense, unless you were prepared to devote a great deal of attention to it. In unpredictable ways, when you did that, it would from time to time reward you.

If you were sufficiently numerate, for example, as we have seen, it would respond to your basic inquiries into its height and base perimeter by 'printing out' the value of pi. And if you were prepared to investigate further, as we shall see, it would download other useful mathematical tit-bits, each a little more complex and abstruse that its predecessor.

There was a programmed feel about this whole process, as though it had been carefully prearranged. Not for the first time, I found myself willing to consider the possibility that the pyramid might have been designed as a gigantic challenge or learning machine - or, better still, as an interactive three-dimensional puzzle set down in the desert for humanity to solve.

Antechamber

Just over 3 feet 6 inches high, the entry passage to the King's Chamber required all humans of normal stature to stoop. About four feet farther on, however, I reached the 'Antechamber', where the roof level rose suddenly to 12 feet above the floor. The east and west walls of the Antechamber were composed of red granite, into which were cut four opposing pairs of wide parallel slots, assumed by Egyptolo- gists to have held thick portcullis slabs: Three of these pairs of slots extended all the way to the floor, and were empty. The fourth (the northernmost) had been cut down only as far as the roof level of the entry passage (that is, 3 feet 6 inches above floor level) and still contained a hulking sheet of granite, perhaps nine inches thick and six feet high. There was a horizontal space of only 21 inches between this suspended stone portcullis and the northern end of the entry passage from which I had just emerged. There was also a gap of a little over 2 feet deep between the top of the portcullis and the ceiling. Whatever

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function it was designed to serve it was hard to agree with the Egyptologists that this peculiar structure could have been intended to deny access to tomb robbers.

Genuinely puzzled, I ducked under it and then stood up again in the southern portion of the Antechamber, which was some 10 feet long and maintained the same roof height of 12 feet. Though much worn, the grooves for the three further 'portcullis' slabs were still visible in the eastern and western walls. There was no sign of the slabs themselves and, indeed, it was difficult to see how such cumbersone pieces of stone could have been installed in so severely constricted a working space.

I remembered that Flinders Petrie, who had systematically surveyed the entire Giza necropolis in the late nineteenth century, had commented on a similar puzzle in the Second Pyramid: 'The granite portcullis in the lower passage shows great skill in moving masses, as it would need 40 or 60 men to lift it; yet it has been moved,

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and raised into place, in a narrow passage, where only a few men could possibly reach it.,3 Exactly the same observations applied to the portcullis slabs of the Great Pyramid. If they were portcullis slabs - gateways capable of being raised and lowered.

The problem was that the physics of raising and lowering them required they be shorter than the full height of the Antechamber, so that they could be drawn into the roof space to allow the entry and exit of legitimate individuals prior to the closure of the tomb. This meant, of course, that when the bottom edges of the slabs were lowered to the floor to block the Antechamber at that level, an equal and opposite space would have opened up between the top edges of the slabs and the ceiling, through which any enterprising tomb-robber would certainly have been able to climb.

The Antechamber clearly qualified as another of the pyramid's many thought-provoking paradoxes, in which complexity of struc- ture was combined with apparent pointlessness of function.

An exit tunnel, the same height and width as the entrance tunnel and lined with solid red granite, led off from the Antechamber's southern wall (also made of granite but incorporating a 12-inch thick limestone layer at its very top). After about a further 9 feet the tunnel debouched into the King's Chamber, a massive sombre red room made entirely of granite, which radiated an atmosphere of prodigious energy and power.

 

Stone enigmas

I moved into the centre of the King's Chamber, the long axis of which was perfectly oriented east to west while the short axis was equally perfectly oriented north to south. The room was exactly 19 feet 1 inch in height and formed a precise two-by-one rectangle measuring 34 feet 4 inches long by 17 feet 2 inches wide. With a floor consisting of 15 massive granite paving stones, and walls composed of 100 gigantic granite blocks, each weighing 70 tons or more and laid in five courses, and with a ceiling spanned by nine further granite blocks each weighing aproximately 50 tons,4 the effect was of intense and overwhelming compression.

At the Chamber's western end was the object which, if the

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Egyptologists were to be believed, the entire Great Pyramid, had been built to house. That object, carved out of one piece of dark chocolate-coloured granite containing peculiarly hard granules of feldspar, quartz and mica, was the lidless coffer presumed to have been the sarcophagus of Khufu.s Its interior measurements were 6 feet 6.6 inches in length, 2 feet 10.42 inches in depth, and 2 feet 2.81 inches in width. Its exterior measurements were 7 feet 5.62 inches in length, 3 feet 5.31 inches in depth, and 3 feet 2.5 inches in width6 an inch too wide, incidentally, for it to have been carried up through the lower (and now plugged) entrance to the ascending corridor.'

Some routine mathematical games were built into the dimensions of the sarcophagus. For example, it had an internal volume of 1 166.4 litres and an external volume of exactly twice that 2332.8 litres.8 Such a precise coincidence could not have been arrived at accidentally: the walls of the coffer had been cut to machine-age tolerances by craftsmen of enormous skill and experience. It seemed, moreover, as Flinders Petrie admitted with some puzzlement after completing his painstaking survey of the Great Pyramid, that these craftsmen had access to tools 'such as we ourselves have only now reinvented. . .'9

Petrie examined the sarcophagus particularly closely and reported that it must have been cut out of its surrounding granite block with straight saws '8 feet or more in length'. Since the granite was extremely hard, he could only assume that these saws must have had bronze blades (the hardest metal then supposedly available) inset with 'cutting points' made of even harder jewels: 'The character of the work would certainly seem to point to diamond as being the cutting jewel; and only the considerations of its rarity in general, and its absence from Egypt, interfere with this conclusion. . .'10

An even bigger mystery surrounded the hollowing out of the sarcophagus, obviously a far more difficult enterprise than separating it from a block of bedrock. Here Petrie concluded that the Egyptians must have:

adapted their sawing principle into a circular instead of a rectilinear form, curving the blade round into a tube, which drilled out a circular groove by its rotation; thus by breaking away the cores left in such grooves, they were able to hollow out large holes with a minimum of

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labour. These tubular drills varied from 1/4 inch to 5 inches diameter, and from 1/30 to 1/5 inch thick.. .11

Of course, as Petrie admitted, no actual jewelled drills or saws had ever been found by Egyptologists. 12 The visible evidence of the kinds of drilling and sawing that had been done, however, compelled him to infer that such instruments must have existed. He became especially interested in this and extended his study to include not only the King's Chamber sarcophagus but many other granite artefacts and granite 'drill cores' which he collected at Giza. The deeper his research, however, the more puzzling the stone-cutting technology of the Ancient Egyptians became:

The amount of pressure, shown by the rapidity with which the drills and saws pierced through the hard stones, is very surprising; probably a load of at least a ton or two was placed on the 4-inch drills cutting in granite. On the granite core No 7 the spiral of the cut sinks 1 inch in the circumference of 6 inches, a rate of ploughing out which is astonishing. . . These rapid spiral grooves cannot be ascribed to anything but the descent of the drill into the granite under enormous pressure. . .13

Wasn't it peculiar that at the supposed dawn of human civilization, more than 4500 years ago, the Ancient Egyptians had acquired what sounded like industrial-age drills packing a ton or more of punch and capable of slicing through hard stones like hot knives through butter?

Petrie could come up with no explanation for this conundrum. Nor was he able to explain the kind of instrument used to cut hieroglyphs into a number of diorite bowls with Fourth Dynasty inscriptions which he found at Giza: 'The hieroglyphs are incised with a very free- cutting point; they are not scraped or ground out, but are ploughed through the diorite, with rough edges to the line. . .'14

This bothered the logical Petrie because he knew that diorite was one of the hardest stones on earth, far harder even than iron.ls Yet here it was in Ancient Egypt being cut with incredible power and precision by some as yet unidentified graving tool:

As the lines are only 1/150 inch wide it is evident that the cutting point rmust have been much harder than quartz; and tough enough not to splinter when so fine an edge was being employed, probably only 1/200

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inch wide. Parallel lines are graved only 1/30 inch apart from centre to centre. 16

In other words, he was envisaging an instrument with a needle-sharp point of exceptional, unprecedented hardness capable of penetrating and furrowing diorite with ease, and capable also of withstanding the enormous pressures required throughout the operation. What sort of instrument was that? By what means would the pressure have been applied? How could sufficient accuracy have been maintained to scour parallel lines at intervals of just 1/30-inch?

At least it was possible to conjure a mental picture of the circular drills with jewelled teeth which Petrie supposed must have been used to hollow out the King's Chamber sarcophagus. I found, however, that it was not so easy to do the same for the unknown instrument capable of incising hieroglyphs into diorite at 2500 BC, at any rate not without assuming the existence of a far higher level of technology than Egyptologists were prepared to consider.

Nor was it just a few hieroglyphs or a few diorite bowls. During my travels in Egypt I had examined many stone vessels - dating back in some cases to pre-dynastic times - that had been mysteriously hollowed out of a range of materials such as diorite, basalt, quartz crystal and metamorphic schist.17

For example, more than 30,000 such vessels had been found in the chambers beneath the Third Dynasty Step Pyramid of Zoser at Saqqara.18 That meant that they were at least as old as Zoser himself (i.e. around 2650 BC I9). Theoretically, they could have been even older than that, because identical vessels had been found in pre-dynastic strata dated to 4000 BC and earlier,20 and because the practice of handing down treasured heirlooms from generation to generation had been deeply ingrained in Egypt since time immemorial.

Whether they were made in 2500 BC or in 4000 BC or even earlier, the stone vessels from the Step Pyramid were remarkable for their workmanship, which once again seemed to have been accomplished by some as yet unimagined (and, indeed, almost unimaginable) tool.

Why unimaginable? Because many of the vessels were tall vases with long, thin, elegant necks and widely flared interiors, often incorporating fully hollowed-out shoulders. No instrument yet invented was capable of carving vases into shapes like these, because

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such an instrument would have had to have been narrow enough to have passed through the necks and strong enough (and of the right shape) to have scoured out the shoulders and the rounded interiors. And how could sufficient upward and outward pressure have been generated and applied within the vases to achieve these effects?

The tall vases were by no means the only enigmatic vessels unearthed from the Pyramid of Zoser, and from a number of other archaic sites. There were monolithic urns with delicate ornamental handles left attached to their exteriors by the carvers. There were bowls, again with extremely narrow necks like the vases, and with widely flared, pot-bellied interiors. There were also open bowls, and almost microscopic vials, and occasional strange wheel-shaped objects cut out of metamorphic schist with inwardly curled edges planed down so fine that they were almost translucent.21 In all cases what was really perplexing was the precision with which the interiors and exteriors of these vessels had been made to correspond - curve matching curve - over absolutely smooth, polished surfaces with no tool marks visible.

There was no technology known to have been available to the Ancient Egyptians capable of achieving such results. Nor, for that matter, would any stone-carver today be able to match them, even if he were working with the best tungsten-carbide tools. The implica-tion, therefore, is that an unknown or secret technology had been put to use in Ancient Egypt.

Ceremony of the sarcophagus

Standing in the King's Chamber, facing west - the direction of death amongst both the Ancient Egyptians and the Maya - I rested my hands lightly on the gnarled granite edge of the sarcophagus which Egyptologists insist had been built to house the body of Khufu. I gazed into its murky depths where the dim electric lighting of the chamber seemed hardly to penetrate and saw specks of dust swirling in a golden cloud.

It was just a trick of light and shadow, of course, but the King's Chamber was full of such illusions. I remembered that Napoleon Bonaparte had paused to spend a night alone here during his conquest

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of Egypt in the late eighteenth century. The next morning he had emerged pale and shaken, having experienced something which had profoundly disturbed him but about which he never afterwards spoke.22

Had he tried to sleep in the sarcophagus?

Acting on impulse, I climbed into the granite coffer and lay down, face upwards, my feet pointed towards the south and my head to the north.

Napoleon was a little guy, so he must have fitted comfortably. There was plenty of room for me too. But had Khufu been here as well?

I relaxed and tried not to worry about the possibility of one of the pyramid guards coming in and finding me in this embarrassing and probably illegal position. Hoping that I would remain undisturbed for a few minutes, I folded my hands across my chest and gave voice to a sustained low-pitched tone - something I had tried out several times before at other points in the King's Chamber. On those occasions, in the centre of the floor, I had noticed that the walls and ceiling seemed to collect the sound, to gather and to amplify it and project it back at me so that I could sense the returning vibrations through my feet and scalp and skin.

Now in the sarcophagus I was aware of very much the same effect, although seemingly amplified and concentrated many times over. It was like being in the sound-box of some giant, resonant musical instrument designed to emit for ever just one reverberating note. The sound was intense and quite disturbing. I imagined it rising out of the coffer and bouncing off the red granite walls and ceiling of the King's Chamber, shooting up through the northern and southern 'ventila-tion' shafts and spreading across the Giza plateau like a sonic mushroom cloud.

With this ambitious vision in my mind, and with the sound of my low-pitched note echoing in my ears and causing the sarcophagus to vibrate around me, I closed my eyes. When I opened them a few minutes later it was to behold a distressing sight: six Japanese tourists of mixed ages and sexes had congregated around the sarcophagus - two of them standing to the east, two to the west and one each to the north and south."

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"... Gathering as much dignity as I could muster, I stood upright smiling and dusting myself off. The Japanese stepped back and I climbed out of the sarcophagus. "