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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