Pyramid Quest (26 page)

The
seked
is essentially a practical rule for architecture and construction, one that delivers a useful value even if you have no idea of the complex geometry that lies behind the concept. Modern construction is full of the same sort of rules. Ask anyone with a general contractor’s license how high and strong the retaining wall has to be if you take 6 feet off the slope of a particular hill, and he or she can give you the right answer even without knowing very much about the theory of gravity or the angle of repose. All you have to know is that dirt has an urge to slide downhill, and it will take a wall of a certain height and strength to stop it. The
seked
had the same sort of value. It told pyramid builders how to angle their structures for greatest esthetic effect, even if they had no idea what pi to six places was or even why that incommensurable number is so central to plane geometry.

Unsurprisingly, Egytologists have seized upon the
seked
theory to explain the slope of the Giza pyramids. Small wonder; it lessens the stature of the Old Kingdom Egyptians as mathematicians but provides a workable explanation for why the Great Pyramid seems to incorporate pi. In effect, the modern
seked
theory says that pi is indeed part of the Great Pyramid, but the ancient Egyptians had no idea what they had stumbled across. The architects of the Old Kingdom had made a lucky find by tinkering, not by doing the math.

Kurt Mendelssohn, a physicist who studied at the University of Berlin under Max Planck and Albert Einstein and did important work in low-temperature physics and the transmutation of elements, also took an interest in the Great Pyramid, offering a variation of this pi-by-practicality theory. Mendelssohn assumes that the ancient Egyptians measured height and distance in different units. Height was stated, of course, in the cubit of 28 fingers. Distance was measured with the rolled cubit. The pyramid builders constructed a drum that was 1 cubit in diameter, then measured the rolled cubit as one revolution of this drum.

Let’s try out the mathematics of Mendelssohn’s theory. The 2-to-1 ratio of height to the length of a side suggests that the intended height of the Great Pyramid was 280 cubits and the length of a single side 140 rolled cubits. The side length would be equal to 140 x π, or 439.8 cubits. In this scenario
h
= 280 cubits and
a
= 70 rolled cubits, so
h
/ a = 280 / (70π) = 4 / π. That is, of course, exactly the value suggested by the pi theory, but again it is found by the Egyptians stumbling upon it rather than understanding the mathematics.

The biggest drawback to Mendelssohn’s theory is lack of evidence. Nothing indicates that the Old Kingdom Egyptians used different units of measure for height and vertical distance or that rolling a drum was their preferred way for determining ground distances. Lack of evidence is also the first of three problems that plague the modern
seked
theory.

Mathematician Roger Herz-Fischler, the one scholar who has investigated the
seked
question, reports that his extensive search of the archaeological literature failed to turn up clear proof that the Egyptians of the Fourth Dynasty used the
seked
as an architectural and construction technique. They could have, but there’s no good evidence to support the contention that they did.

In the absence of evidence, contemporary Egyptologists who argue the
seked
theory are making the assumption that what was true of Twelfth Dynasty Egyptians must have been true for Fourth Dynasty Egyptians. They are dismissing as irrelevant the severe decline in cultural and intellectual skills that accompanied the collapse of the Old Kingdom and the eventual rise of the Middle Kingdom from political anarchy and social chaos. Compare the buildings of the Middle Kingdom with those of the Old, and you see a marked decline in esthetics and construction. The newer structures are like poorly done sketches of masterworks. The same kind of upheaval and decline probably occurred in Egyptian intellectual life.

Remember what happened in Europe in late antiquity and the medieval period. Although the artistic and intellectual accomplishments of Greece and Rome now seem like a fundamental part of European civilization, Europe completely lost contact with the Greek and Roman classics throughout the Dark Ages, which followed the collapse of the Roman Empire in the fifth century A.D. If it hadn’t been for contact during the Crusades with Arab scholars who continued to study Greek writers and the reintroduction of classic Latin texts from Irish monasteries that had faithfully preserved and copied them for centuries, modern Europeans still wouldn’t know who Cicero or Aristotle was.

There is every chance that something of the same occurred in Egypt. The Middle Kingdom Egyptians may well have lost the pi the Old Kingdom Egyptians had and settled for the
seked
as a reasonable facsimile.

Which leads us to the final problem with the
seked
theory: the intriguing suggestion that pi wasn’t the only mathematical constant the Old Kingdom Egyptians understood.

Since the Renaissance it has been called the Golden Section, or phi (Φ). Phi isn’t a number that can be worked out with arithmetic, but it can be determined with nothing more than a compass and a ruler. First, draw a line, which we call AC. Now divide the line AC at a point B such that AC / AB = AB / BC. In other words, the ratio of whole to the longer part is the same as the ratio of the longer part to the shorter part. Both ratios equal phi, which is 1.618033988749895 . . . This irrational and incommensurable number is the Golden Section, also known as the Golden Mean, Primordial Scission, and the Divine Proportion.

Phi can also be demonstrated with the geometry of the square. Take a square with a side of 1 unit, and cut it in half from one side to the other, forming two rectangles of 1 x (½). The diagonal of one of the rectangles plus ½ 2 equals Φ. Let’s call this diagonal
W,
and apply the Pythagorean theorem to it. Now we know the relationship of
W
to the other two sides:
W
²
= 1
²
+ (1/2)
²
. This can also be written as W
²
= 1.25, so W = √1.25 and Φ = √1.25 + (1/2). However, √1.25 can be multiplied by 1 in the form of √4 / 2 to arrive at √4 x1 .25/2 = √5 / 2. Now substitute √5 / 2 for √1.25 in the equation Φ = √1.25 + ½, and we arrive at Φ = (1 + √5) / 2.

One of the most fascinating things about phi is that 1 + Φ = Φ
²
. Carry out some simple algebra on this equation, and you get (1 / Φ) + 1 = Φ, an equation that leads to an additive series of numbers known as the Fibonacci series. It is named for perhaps the greatest mathematical genius of the Middle Ages, Leonardo Fibonacci (c. 1270-1240 A.D.), an Italian also known as Leonardo of Pisa. Fibonacci introduced Europe to the Hindu-Arabic numbers we know and use, and he traveled widely, even to Egypt, and studied the mathematical techniques he encountered. It may have been in Egypt that Fibonacci first encountered the sequence that bears his name and discovered its relationship to both phi and pi.

The Fibonacci sequence looks simple enough: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 . . . Each number after the first 1 represents the sum of the two preceding numbers. Intriguingly, the ratio of each term to the one that goes before yields an approximation of phi. As you go up the series, the approximation gets more and more accurate. The ratio of 1:1 is 1, 3:2 is 1.5, 5:3 is 1.666, and by the time you work up the series to 55:34, the ratio is 1.61747, much closer to the actual value of 1.6180339.

Through the Fibonacci series, phi shapes many natural phenomena, such as the growth curve of a nautilus shell, the seed whorls in a sunflower or aster, and the structure of a spiral galaxy. Plato in his dialogue
Timaeus
—which also discusses Atlantis—says that the Golden Section is the most binding of all mathematical relationships and that it is a key to the physics of the cosmos. The Golden Section is also an important compositional element in many Renaissance paintings, including works by Fra Lippo Lippi (1406-1469), Leonardo da Vinci (1452-1519), and Raphael (1483-1520). It formed the basis of the grid system used by Le Corbusier (1887-1965), the great Swiss architect who designed, among other buildings, the United Nations headquarters in New York City.

The classical Athenians used the Golden Section in constructing the Acropolis, and the elaboration of the mathematics behind it is associated with the Greek geometers Pythagoras (c. 569-475 B.C.) and Euclid (c. 325- 265 B.C.). But the Great Pyramid and other monuments suggest that the Egyptians of the Old Kingdom understood phi, and its relationship to pi, more than 2,000 years earlier.

Perhaps the first writer to make this assertion was René Schwaller de Lubicz (1887-1961), the Alsatian mathematician and philosopher whose observation of water erosion on the Great Sphinx was the roundabout reason I first came to Giza. Consider, for example, a relief Schwaller studied on the east side of the temple of Luxor, which received more of his attention than any other ancient Egyptian structure. The relief shows a group of priests carrying the solar barque of the king through a gate in the temple of Karnak.
²
According to Schwaller’s computations, if the width of the gate from one outside wall to another is taken as 1, the external height of the gate is 2, while if the width of the gate from one inside wall to the other is taken as 1, the height of the inside of the gate is Φ
²
× 1.2 = 3.1416.

That’s the value of pi, of course, and it means that the ancient Egyptians understood the relationship of pi to phi as π = Φ
²
× 6 / 5. Take two approximations of Φ in the Fibonacci series in sequence and substitute then into this equation, and you can produce a good approximation of pi (the pi approximations, like the phi approximations, get closer as you move further along in the Fibonacci series). This gives us at least one approximation of pi apparently used in the Great Pyramid, namely (34 / 21) × (55 / 34) × (6 / 5) = (55 / 21) × (6 / 5) = (11/21) × 6 = 66 / 21 = 22 / 7.

To Schwaller’s credit, he based his revelation of phi in Old Kingdom Egypt on more than one observation. Many depictions of the pharaohs show the Egyptian king wearing a curious kind of triangular loincloth. Schwaller de Lubicz measured the angles of dozens upon dozens of these depictions and always came up with two values: Φ and √φ. It is no sym bolic accident that the loincloth was the article of clothing chosen to represent phi. Because of phi’s importance in shaping everything in the world from nautilus shells to spiral galaxies, the number is commonly seen as the seed power of the universe. Phi is phallic.

Schwaller also argued that phi appears in the cross-section of the Great Pyramid, which forms a triangle comprising the height of the structure, half its base, and the apothem. If the half-base is 1, then the apothem is Φ, and the height is √φ. The cross-section of the Great Pyramid, therefore, expresses the same angles as the pharaohs’ loincloths and embodies the same masculine principle of the shape-forming seed.

Livio Catullo Stecchini, the measures-obsessed classicist we met in chapter 7, added an additional wrinkle to the presence of phi in the Great Pyramid and its relationship to pi. Most Great Pyramid researchers assume that the structure was intended to have a perfectly square base with sides that rose to the apex at equal angles. Stecchini questioned these fundamental assumptions. He believed that a starting point for the design of the Great Pyramid may have been a base length of 440 cubits and a height of 280 cubits, but that these dimensions were then modified in the final plan and construction. The basic length of each side was changed to 439.5 cubits, and the perimeter of the Great Pyramid was therefore meant to be 1,758 cubits (921.453 meters). You will remember from chapter 7 that Stecchini said that this value was meant to equal a half-minute of latitude at the equator. The ancient Egyptians calculated this value as 3,516 cubits, which translates to 1,842.905 meters, extremely close to our contemporary measurement of 1,842.925 meters.

But Stecchini goes further. The Cole Survey of 1925 shows that the Great Pyramid isn’t quite perfectly square. Most Egyptologists ascribe this slight variation to accident or inaccuracy. After all, it’s difficult to put that much stone into place and get it all right down to the last cubit or two. Stecchini, however, says that the Great Pyramid’s base was designed to be slightly different from a perfect square, and that the purpose behind this variation has to do with pi and phi.

As Stecchini sees it, the axis for the alignment of the Great Pyramid’s western side was laid out first, then the northern side was drawn to be perfectly perpendicular to it. The eastern side, however, was intentionally positioned at an angle 3 arcminutes greater than perpendicular to the northern side. In other words, the northeast corner was meant to be 90° 03’ 00”, not 90°. As for the southern side, it was intended to be a half-arcminute greater than perpendicular, so that the southwest corner measured 90° 00’ 30”, for according to Stecchini’s analysis, not all of the sides were at the exact same angles.

Stecchini also analyzed a small line on the pavement at the base of the Great Pyramid near the middle on the north side. Some writers have assumed that this was the original north-south axis of the Great Pyramid. The data from the Cole Survey show that the axis line is located 115.090 meters from the northwest corner and 115.161 meters from the northeast corner, so that it is a little off center, a variation typically dismissed as human error. Stecchini concludes that this was no mistake. Rather, the Great Pyramid’s north-south axis was off center on purpose. Therefore, the apex was also off center—again, on purpose—by about 35.5 millimeters to the west. As a result, each of the four faces of the Great Pyramid has a slightly different slope from the others, an idea that also occurred to Petrie in the course of his measurements but one he never pursued.