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  (10 inches x 2 x 3.14) and a circle with a radius of 7 inches will have a

  circumference of 43.96 inches (7 inches x 2 x 3.14).

  These formulae using the value of pi for calculating circumference from

  either diameter or radius apply to all circles, no matter how large or how

  small, and also, of course, to all spheres and hemispheres. They seem

  relatively simple—with hindsight. Yet their discovery, which represented a

  revolutionary breakthrough in mathematics, is thought to have been

  made late in human history. The orthodox view is that Archimedes in the

  third century BC was the first man to calculate pi correctly at 3.14.8

  Scholars do not accept that any of the mathematicians of the New World

  ever got anywhere near pi before the arrival of the Europeans in the

  sixteenth century. It is therefore disorienting to discover that the Great

  Pyramid at Giza (built more than 2000 years before the birth of

  Archimedes) and the Pyramid of the Sun at Teotihuacan, which vastly

  predates the conquest, both incorporate the value of pi. They do so,

  moreover, in much the same way, and in a manner which leaves no doubt

  that the ancient builders on both sides of the Atlantic were thoroughly

  conversant with this transcendental number.

  The principal factors involved in the geometry of any pyramid are (1)

  the height of the summit above the ground, and (2) the perimeter of the

  monument at ground level. Where the Great Pyramid is concerned, the

  ratio between the original height (481.3949 feet9) and the perimeter

  (3023.16 feet10) turns out to be the same as the ratio between the radius

  and the circumference of a circle, i.e. 2pi.11 Thus, if we take the pyramid’s

  height and multiply it by 2pi (as we would with a circle’s radius to

  calculate its circumference) we get an accurate read-out of the

  monument’s perimeter (481.3949 feet 2 x 3.14 = 3023.16 feet).

  Alternatively, if we turn the equation around and start with the

  circumference at ground level, we get an equally accurate read-out of the

  height of the summit (3023.16 feet divided by 2 divided by 3.14 =

  481.3949 feet).

  Since it is almost inconceivable that such a precise mathematical

  correlation could have come about by chance, we are obliged to conclude

  that the builders of the Great Pyramid were indeed conversant with pi and

  that they deliberately incorporated its value into the dimensions of their

  monument.

  Now let us consider the Pyramid of the Sun at Teotihuacan. The angle of

  its sides is 43.5°12 (as opposed to 52° in the case of the Great Pyramid13).

  The Mexican monument has the gentler slope because the perimeter of

  8 Encyclopaedia Britannica, 9:415.

  9 I. E. S. Edwards, The Pyramids of Egypt, Penguin, London, 1949, p. 87.

  10 Ibid.

  11 Ibid., p. 219.

  12 Mysteries of the Mexican Pyramids, p. 55.

  13 The Pyramids of Egypt, pp. 87, 219.

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  its base, at 2932.8 feet,14 is not much smaller than that of its Egyptian

  counterpart while its summit is considerably lower (approximately 233.5

  feet prior to Bartres’s, ‘restoration’15).

  The 2pi formula that worked at the Great Pyramid does not work with

  these measurements. A 4pi formula does. Thus if we take the height of

  the Pyramid of the Sun (233.5 feet) and multiply it by 4pi we once again

  obtain a very accurate read-out of the perimeter: 233.5 feet x 4 x 3.14 =

  2932.76 feet (a discrepancy of less than half an inch from the true figure

  of 2932.8 feet).

  This, surely, can no more be a coincidence than the pi relationship

  extrapolated from the dimensions of the Egyptian monument. Moreover,

  the very fact that both structures incorporate pi relationships (when none

  of the other pyramids on either side of the Atlantic does) strongly

  suggests not only the existence of advanced mathematical knowledge in

  antiquity but some sort of underlying common purpose.

  The height of the Pyramid of the Sun x 4pi = the perimeter of its

  base. The height of the Great Pyramid at Giza x 2 pi = the perimeter of

  its base.

  As we have seen the desired height/perimeter ratio of the Great

  14 The Ancient Kingdoms of Mexico, p. 74.

  15 Mexico, p. 201; The Atlas of Mysterious Places, p. 156.

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  Pyramid ( 2pi) called for the specification of a tricky and idiosyncratic

  angle of slope for its sides: 52°. Likewise, the desired height/perimeter

  ratio of the Pyramid of the Sun ( 4pi) called for the specification of an

  equally eccentric angle of slope: 43.5°. If there had been no ulterior

  motive, it would surely have been simpler for the Ancient Egyptian and

  Mexican architects to have opted for 45° (which they could easily have

  obtained and checked by bisecting a right angle).

  What could have been the common purpose that led the pyramid

  builders on both sides of the Atlantic to such lengths to structure the

  value of pi so precisely into these two remarkable monuments? Since

  there seems to have been no direct contact between the civilizations of

  Mexico and Egypt in the periods when the pyramids were built, is it not

  reasonable to deduce that both, at some remote date, inherited certain

  ideas from a common source?

  Is it possible that the shared idea expressed in the Great Pyramid and

  the Pyramid of the Sun could have to do with spheres, since these, like

  the pyramids, are three-dimensional objects (while circles, for example,

  have only two dimensions)? The desire to symbolize spheres in threedimensional monuments with flat surfaces would explain why so much

  trouble was taken to ensure that both incorporated unmistakable pi

  relationships. Furthermore it seems likely that the intention of the

  builders of both of these monuments was not to symbolize spheres in

  general but to focus attention on one sphere in particular: the planet

  earth.

  It will be a long while before orthodox archaeologists are prepared to

  accept that some peoples of the ancient world were advanced enough in

  science to have possessed good information about the shape and size of

  the earth. However, according to the calculations of Livio Catullo

  Stecchini, an American professor of the History of Science and an

  acknowledged expert on ancient measurement, the evidence for the

  existence of such anomalous knowledge in antiquity is irrefutable.16

  Stecchini’s conclusions, which relate mainly to Egypt, are particularly

  impressive because they are drawn from mathematical and astronomical

  data which, by common consent, are beyond serious dispute.17 A fuller

  examination of these conclusions, and of the nature of the data on which

  they rest, is presented in Part VII. At this point, however, a few words

  from Stecchini may shed further light on the mystery that confronts us:

  The basic idea of the Great Pyramid was that it should be a representation of the

  northern hemisphere of the earth, a hemisphere projected on flat-surfaces as is

 
done in map-making ... The Great Pyramid was a projection on four triangular

  surfaces. The apex represented the pole and the perimeter represented the

  equator. This is the reason why the perimeter is in relation 2pi to the height. The

  16 The most accessible presentation of Stecchini’s work is in the appendix he wrote for

  Peter Tompkins, Secrets of the Great Pyramid, pp. 287-382.

  17 See The Traveller’s Key to Ancient Egypt, p. 95.

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  Great Pyramid represents the northern hemisphere in a scale of 1:43,200.18

  In Part VII we shall see why this scale was chosen.

  Mathematical city

  Rising up ahead of me as I walked towards the northern end of the Street

  of the Dead, the Pyramid of the Moon, mercifully undamaged by

  restorers, had kept its original form as a four-stage ziggurat. The Pyramid

  of the Sun, too, had consisted of four stages but Bartres had whimsically

  sculpted in a fifth stage between the original third and fourth levels.

  There was, however, one original feature of the Pyramid of the Sun that

  Bartres had been unable to despoil: a subterranean passageway leading

  from a natural cave under the west face. After its accidental discovery in

  1971 this passageway was thoroughly explored. Seven feet high, it was

  found to run eastwards for more than 300 feet until it reached a point

  close to the pyramid’s geometrical centre.19 Here it debouched into a

  second cave, of spacious dimensions, which had been artificially enlarged

  into a shape very similar to that of a four-leaf clover. The ‘leaves’ were

  chambers, each about sixty feet in circumference, containing a variety of

  artefacts such as beautifully engraved slate discs and highly polished

  mirrors. There was also a complex drainage system of interlocking

  segments of carved rock pipes.20

  This last feature was particularly puzzling because there was no known

  source of water within the pyramid.21 The sluices, however, left little

  doubt that water must have been present in antiquity, most probably in

  large quantities. This brought to mind the evidence for water having once

  run in the Street of the Dead, the sluices and partition walls I had seen

  earlier to the north of the Citadel, and Schlemmer’s theory of reflecting

  pools and seismic forecasting.

  Indeed, the more I thought about it the more it seemed that water had

  been the dominant motif at Teotihuacan. Though I had hardly registered

  it that morning, the Temple of Quetzalcoatl had been decorated not only

  with effigies of the Plumed Serpent but with unmistakable aquatic

  symbolism, notably an undulating design suggestive of waves and large

  numbers of beautiful carvings of seashells. With these images in my

  mind, I reached the wide plaza at the base of the Pyramid of the Moon

  and imagined it filled with water, as it might have been, to a depth of

  about ten feet. It would have looked magnificent: majestic, powerful and

  18 Stecchini, in appendix to Secrets of the Great Pyramid, p. 378. The perimeter of the

  Great Pyramid equals exactly one-half minute of arc—see Mysteries of the Mexican

  Pyramids, p. 279.

  19 The Pyramids of Teotihuacan, p. 20.

  20 Mysteries of the Mexican Pyramids, pp. 335-9.

  21 Ibid.

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  serene.

  The Akapana Pyramid in far-off Tiahuanaco had also been surrounded

  by water, which had been the dominant motif there—just as I now found

  it to be at Teotihuacan.

  I began to climb the Pyramid of the Moon. It was smaller than the

  Pyramid of the Sun, indeed less than half the size, and was estimated to

  be made up of about one million tons of stone and earth, as against two

  and a half million tons in the case of the Pyramid of the Sun. The two

  monuments, in other words, had a combined weight of three and a half

  million tons. It was thought unlikely that this quantity of material could

  have been manipulated by fewer than 15,000 men and it was calculated

  that such a workforce would have taken at least thirty years to complete

  such an enormous task.22

  Sufficient labourers would certainly have been available in the vicinity:

  the Teotihuacan Mapping Project had demonstrated that the population

  of the city in its heyday could have been as large as 200,000, making it a

  bigger metropolis than Imperial Rome of the Caesars. The Project had

  also established that the main monuments visible today covered just a

  small part of the overall area of ancient Teotihuacan. At its peak the city

  had extended across more than twelve square miles and had

  incorporated some 50,000 individual dwellings in 2000 apartment

  compounds, 600 subsidiary pyramids and temples, and 500 ‘factory’

  areas specializing in ceramic, figurine, lapidary, shell, basalt, slate and

  ground-stone work.23

  At the top level of the Pyramid of the Moon I paused and turned slowly

  around. Across the valley floor, which sloped gently downhill to the

  south, the whole of Teotihuacan now stretched before me—a geometrical

  city, designed and built by unknown architects in the time before history

  began. In the east, overlooking the arrow-straight Street of the Dead,

  loomed the Pyramid of the Sun, eternally ‘printing out’ the mathematical

  message it had been programmed with long ages ago, a message which

  seemed to direct our attention to the shape of the earth. It almost looked

  as though the civilization that had built Teotihuacan had made a

  deliberate choice to encode complex information in enduring monuments

  and to do it using a mathematical language.

  Why a mathematical language?

  Perhaps because, no matter what extreme changes and transformations

  human civilization might go through, the radius of a circle multiplied by

  2pi (or half the radius multiplied by 4pi) would always give the correct

  figure for that circle’s circumference. In other words, a mathematical

  language could have been chosen for practical reasons: unlike any verbal

  tongue, such a code could always be deciphered, even by people from

  22 The Riddle of the Pyramids, pp. 188-93.

  23 The Prehistory of the Americas, p. 281. See also The Cities of Ancient Mexico, p. 178

  and Mysteries of the Mexican Pyramids, pp. 226-36.

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  unrelated cultures living thousands of years in the future.

  Not for the first time I felt myself confronted by the dizzying possibility

  that an entire episode in the story of mankind might have been forgotten.

  Indeed it seemed to me then, as I overlooked the mathematical city of the

  gods from the summit of the Pyramid of the Moon, that our species could

  have been afflicted with some terrible amnesia and that the dark period

  so blithely and dismissively referred to as ‘prehistory’ might turn out to

  conceal unimagined truths about our own past.

  What is prehistory, after all, if not a time forgotten—a time for which we

  have no records? What is prehistory if not an epoch of impenetrable

  obscurity through which our ancestors passed but a
bout which we have

  no conscious remembrance? It was out of this epoch of obscurity,

  configured in mathematical code along astronomical and geodetic lines,

  that Teotihuacan with all its riddles was sent down to us. And out of that

  same epoch came the great Olmec sculptures, the inexplicably precise

  and accurate calendar the Mayans inherited from their predecessors, the

  inscrutable geoglyphs of Nazca, the mysterious Andean city of

  Tiahuanaco ... and so many other marvels of which we do not know the

  provenance.

  It is almost as though we have awakened into the daylight of history

  from a long and troubled sleep, and yet continue to be disturbed by the

  faint but haunting echoes of our dreams ...

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  Part IV

  The Mystery of the Myths

  1. A Species with Amnesia

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  Chapter 24

  Echoes of Our Dreams

  In some of the most powerful and enduring myths that we have inherited

  from ancient times, our species seems to have retained a confused but

  resonant memory of a terrifying global catastrophe.

  Where do these myths come from?

  Why, though they derive from unrelated cultures, are their storylines so

  similar? why are they laden with common symbolism? and why do they so

  often share the same stock characters and plots? If they are indeed

  memories, why are there no historical records of the planetary disaster

  they seem to refer to?

  Could it be that the myths themselves are historical records? Could it be

  that these cunning and immortal stories, composed by anonymous

  geniuses, were the medium used to record such information and pass it

  on in the time before history began?

  And the ark went upon the face of the waters

  There was a king, in ancient Sumer, who sought eternal life. His name

  was Gilgamesh. We know of his exploits because the myths and traditions

  of Mesopotamia, inscribed in cuneiform script upon tablets of baked clay,

  have survived. Many thousands of these tablets, some dating back to the

  beginning of the third millennium BC, have been excavated from the

  sands of modern Iraq. They transmit a unique picture of a vanished

  culture and remind us that even in those days of lofty antiquity human

  beings preserved memories of times still more remote—times from which