Pyramid Quest (25 page)

We all share an urge to see our time as the most critical historical period ever. That urge reveals itself powerfully in the prophetic writing about the Great Pyramid. This came out almost ironically in a used copy of Lemesurier’s book I was reading. A previous owner of the volume had underlined a passage where Lemesurier wrote: “both Bible and Pyramid agree that the second advent will be preceded by a time of unprecedented death and destruction”
²²
and penned “AIDS?” in the margin. Think about this. The centuries since the building of the Great Pyramid and the writing of the Old and News Testaments have seen far too many examples of death and destruction on an unprecedented scale. Consider the Black Death, the religious wars of medieval and early modern Europe, the 1918 influenza epidemic, the European invasion of the New World, the 20 million Russian dead in World War II, the fire-bombing of Tokyo and Dresden, the Holocaust, Hiroshima and Nagasaki, the killing fields of Cambodia, the Rwanda genocide—the list goes on and on. But that unknown reader who came before me wrote AIDS, because this terrible disease is the prime contemporary example of widespread death. The issue is less the horror of the epidemic than the fact that it is
our
epidemic.

And that is the failure of the prophetic writing about the Great Pyramid. It tries to appropriate this amazing structure to a religious perspective the monument itself does not share. The challenge in understanding the Great Pyramid is taking it on its own terms, as an achievement of the third millennium B.C. in the Old Kingdom, and not on the basis of the modern prejudices and ideologies we bring to it.

But we have to give the prophetic writing credit for one important understanding. From Greaves to Lemesurier, this tradition shows that the numbers in the Great Pyramid matter—but only when we attempt to understand them as the great builders did. This brings us the question of pi and the Golden Section.

PI AND THE GOLDEN SECTION

TO CHRISTIAN FUNDAMENTALIST AND PYRAMID SCHOLAR John Taylor, one of the numbers that counted toward his proof of the Great Pyramid’s divine origin was pi (π). Pi expresses the relationship between the circumference of a circle and its diameter. If
C
is the circumference and
d
is the diameter, then
C
/
d
= π. Since the radius (
r
) of a circle equals half the diameter, or 2
r
=
d,
this equation can also be written as
C
/ (2
r
) = π.

The beauty of pi is that it allows you to calculate the circumference of a circle from the measurement of the radius. Begin with the
C
/ (2
r
) = π equation, then go through the usual algebraic operations to solve for
C.
This makes the equation for calculating the circumference from the radius 2π
r
=
C.

Pi has another important and fascinating mathematical characteristic: it is an incommensurable number. Take it out to as many places as you want—modern computers have performed the calculation to hundreds of thousands of places—and it just keeps going on, and on, and on. Stop anywhere in that lengthy calculation, no matter how far out, and the number you end up with is in some manner an approximation. For practical reasons, modern mathematicians use 3.14159+ as the best six-digit value for pi (= 3.14159265358979 . . . ).

According to the standard history of science, the Babylonians were the first to discover that pi exists. They also worked out the first approximation, 3.125, in about 2000 B.C. It was another 1,700 years before Archimedes of Syracuse (287?-212 B.C.), by many reckonings the most brilliant mathematician of antiquity, found a method for determining pi to any desired accuracy. In the fifth century A.D., the Chinese mathematician Tsu Chung-Chi determined that the exact value of pi was both greater than 3.1415926 and less than 3.1415927, a degree of accuracy that wasn’t achieved in Europe until the sixteenth century.

Given this history, John Taylor was surprised when he played with numbers from the Great Pyramid and discovered what he thought just had to be pi, long before anyone was supposed to have known it existed.

Even though Taylor took credit for the pi theory, it almost certainly didn’t originate with him. He probably borrowed it from another much more obscure writer, H. Agnew, who in 1838 published a book with the delightful title
Letter from Alexandria on the Practical Application of the Quadrature of the Circle in the Configuration of the Great Pyramids of Egypt.
Little is known of Agnew except that he spent the year 1835 in Cairo, which was under quarantine because of an outbreak of the plague. Rather than idle about, Agnew used the time to analyze the Giza pyramids, which were conveniently close to the shut-down city. By the time he finished, Agnew decided that the mathematical ratio between the Menkaure Pyramid’s height and its perimeter depended on the constant pi.

John Taylor never credited Agnew with this insight, but he appeared to make use of it in his study of the Great Pyramid (Agnew’s ideas are discussed in Vyse, 1840, which Taylor had read). He discovered that dividing the perimeter of the Great Pyramid by twice its height gave 3.144 as the quotient. Since this value lies in the same neighborhood as 3.14159+ (if rounded to 3.142, pi is only 0.002 from Taylor’s quotient), and since the pyramid’s height had the same relationship to its perimeter that a circle’s circumference has to its radius, Taylor argued that the intended value was pi—perhaps in the then-current Egyptian approximation. He argued further that the builders had incorporated pi into the pyramid by design. In fact, Taylor saw the emergence of pi, centuries before its putative discovery, as further evidence that God, and not a raggedy bunch of idol-worshipping Egyptians, provided the inspiration for the Great Pyramid. Taylor’s most enthusiastic follower was Charles Piazzi Smyth, who, like Taylor, seized upon the pi theory as further proof that the Great Pyramid was God’s handiwork.

Against the backdrop of Victorian England, such a claim sounded more plausible than it might now. Unlike the French or the Germans, the English saw mathematics as both central to a liberal education and woven from the same fabric as theology. English gentlemen typically received at least a basic education in mathematics, and they were of a mind to look for proofs of divinity within it. In their world, the existence and nature of God were seen as immutable as geometric truth. Science gave humans an insight into the nature of the divine; geometry, like all science, functioned not only as a standard of immutability but also as a direct path to God. John Henry Cardinal Newman (1801-1890), who was both a leading Victorian intellectual and a prominent Roman Catholic churchman, wrote that “religious doctrine is knowledge, in as full a sense as Newton’s doctrine is knowledge.” He even alluded to John 3:16, a gospel verse much beloved by fundamentalist Christians, when he wrote “God so loved the world that He made it good, and gave man a mind with which to investigate and display God’s goodness in the form known as knowledge, scientific knowledge, even technical knowledge.”
¹

As a result, Taylor’s and Smyth’s ideas received a careful hearing, and they circulated far outside Christian and fundamentalist circles in the English-speaking world. They gave rise to the pi theory of the Great Pyramid, which to this day attracts a great deal of attention. Despite their flimsy evidence for their prophetic theory of pyramid building, Taylor and Smyth may have made an important observation. Sometimes people with the wrong models come up with the right theories, and the pi theory deserves a close look.

It goes like this: the pyramid’s shape was determined by, first, setting its height to equal the radius of a hypothetical circle, then setting its base perimeter to equal the circumference of the same circle. Assuming that all four of the pyramid’s sides are of equal length, each side equals one-quarter of the circumference of the same hypothetical circle. This set of relationships determined the pyramid’s slope.

The calculation is simple and direct. Let
L
be the length of one equal side and let
h
be the height of the pyramid. Therefore, 2
h
π = 4
L,
or π = 2
L
/
h .
Now let
a
stand for the horizontal distance from the middle of one side of the Great Pyramid to a point directly below the apex; then 2
a
=
L.
Next, substituting the value 2
a
for
L
into the first equation gives π = 4
a
/
h .
The tangent of the slope of the Great Pyramid can be determined by rearranging this equation to give
h
/
a
= 4 / π
.

The equation that summarizes the pi theory,
h
/
a
= 4 / π, gives the run relative to the rise (or rise over run). That is, it tells how much the pyramid had to rise, a measure of vertical displacement, for every given unit of run, a measure of horizontal displacement. The equation tells us that for every rise of 4, the pyramid had to show a run of π. This relationship works well on the blackboard or in an electronic calculator, but 3.14159+ is a tough value to use in a quarry for cutting and measuring blocks, or on a hot, dusty building site teeming with work gangs. Whole numbers are a lot easier to handle. The units you use for those numbers, whether cubits or pyramid inches or meters, do not matter, since they cancel out mathematically. As a result, the Egyptians may have settled for 22/7 as a practical value for pi, even if it is not quite so accurate as 3.14159+. If they used 22/7 as a convenient approximation for π, then
h
/
a
= 4 / π = 4 / (22 / 7) = 28 / 22 = 14 / 11. All the Egyptians had to do was to set a rise of 14 units for every horizontal run of 11 units, and they were building a pi-theory pyramid.

Of course, the rise-to-run ratio set the slope of the pyramid. If the modern approximation of pi was incorporated into the Great Pyramid, the slope would be 51.854°. But if it was 22/7, the slope would be approximately 51.843°. The difference is small, but it is a difference. Theoretically, if we can just measure the slope of the Great Pyramid, we can determine how close it is to the modern or 22/7 approximation of pi, and then decide on the basis of the evidence whether the Old Kingdom Egyptians knew what pi was all about. The trouble is, the reality of the Great Pyramid is less than cooperative.

Since the original casing stones have been removed from the Great Pyramid and we are looking at the imprecise core, we don’t know with real accuracy what the original slope was. Sir Flinders Petrie (1853-1942), whose methodical archaeological studies of the Great Pyramid were motivated by a desire to test the biblically based theories of Taylor and Smyth, calculated slopes based on the few remaining casing stones on the north face and one on the south. On the north side, Petrie came up with values that ranged from 51.736° to 51.889°, while the south face had a slope of 51.958°. From this fairly meager evidence, Petrie concluded that the mean slope of the Great Pyramid was 51.866°.

That value is only 0.012° off the 51.854° slope for the modern approximation of pi and 0.023 ° from the 51.843° slope for the 22/7 value. Unfortunately, these results doesn’t prove very much, because there is another way the Egyptians could have gotten to the same value, even without having a handle on pi.

In 1858 Alexander Henry Rhind, a Scottish antiquary traveling in Egypt, purchased a papyrus that has proved to be one of the oldest mathematical documents known. Today, because of its historical importance, that papyrus resides in the British Museum. Dated to approximately 1550 B.C. in the Fifteenth Dynasty, or about one millennium after the construction of the Great Pyramid, the Rhind Papyrus is also known as the Ahmes (or Ahmose) Papyrus, after the name of the scribe who copied it from an original that was about 300 years older. The Rhind Papyrus gives insight into the nature of mathematics in the latter half of the Twelfth Dynasty, about 700 years after the completion of the Great Pyramid.

The Rhind Papyrus is what we would call a manual of mathematical techniques. It shows, through numerous examples, how to solve basic kinds of problems in arithmetic and geometry. A great deal of variation exists in the text, so it may show techniques developed at different historical times.

Certain sample problems in the papyrus dealing with pyramids incorporate a concept known as the
seked.
The
seked
measures the run of the pyramid relative to a rise of one cubit, the Egyptian measure we encountered in chapter 8. One cubit consists of 7 palms, each of which is divided into 4 fingers; 1 cubit, therefore, equals 28 fingers. According to the concept of the
seked
in the Rhind Papyrus, producing a pyramid with the approximate angle of the Great Pyramid at Giza requires a run of 5 palms and 2 fingers for every cubit of rise. In other words, for every 28 fingers the pyramid rose vertically, it had to extend 22 fingers horizontally. Mathematically, this is
seked:
28 / 22 = 14 / 11. This is exactly the same run-to-rise ratio that comes from a 22/7 approximation of pi.