The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)
3. UNDER A STAR-Y-POINTING PYRAMID?
The Pyramids first, which in Egypt were laid;
Next Babylon's Gardens, for Amytis made;
Then Mausolos' Tomb of affection and guilt;
Fourth, the Temple of Diana in Ephesus built;
The Colossus of Rhodes, cast in brass, to the
Sun; Sixth, Jupiter's Statue, by Phidias done;
The Pharos of Egypt comes last, we are told,
Or the Palace of Cyrus, cemented with gold.
—ANONYMOUS, “SEVEN WONDERS OF THE ANCIENT WORLD”
The title of this chapter comes from the poem On Shakespeare, written in 1630 by the famous English poet John Milton (1608-1674). Milton, who himself was widely esteemed as a poet second only to Shakespeare, writes:
What needs my Shakespeare for his honor'd bones
The labor of an age in piled stones?
Or that his hallow'd relics should be hid
Under a star-y-pointing pyramid?
Dear son of memory, great heir of fame,
What need'st thou such weak witness of thy name?
As we shall soon see, the alignment of the pyramids was indeed based on the stars. As if these monuments are not awe-inspiring enough, however, many authors insist that the Great Pyramid's dimensions are based on the Golden Ratio. To all Golden Ratio enthusiasts, this association only adds to the air of mystique surrounding this proportion. But is this true? Did the ancient Egyptians really know about ö, and if they did, did they truly choose to “immortalize” the Golden Ratio by incorporating it into one of the Seven Wonders of the Ancient World?
Seeing that the initial interest in the Golden Ratio probably was inspired by its relation to the pentagram, we must first follow some of the early history of the pentagram, since this may lead us to the earliest occurrences of the Golden Ratio.
Ask any child to draw you a star and she will most likely draw a pentagram. This is actually a consequence of the fact that we happen to view stars through Earth's atmosphere. The turbulence of the air bends starlight in constantly shifting patterns, thus causing the familiar twinkling. In an attempt to represent the spikes generated by twinkling using a simple geometrical shape, humans came up with the pentagram, which also has the additional attractive property that it can be drawn without lifting the writing tool off the clay, papyrus, or paper.
Over the ages, such “stars” have become a symbol of excellence (e.g., five-star hotels, movies, book reviews), achievement (“stardom”), opportunity (“reach for the stars”), and authority (“five-star” generals). When this symbolism is combined with the romantic appeal of a starry night, it is no wonder that the flags of more than sixty nations depict five-pointed stars and that such star patterns appear on innumerable commercial logos (e.g., Texaco, Chrysler).
Some of the earliest known pentagrams come from fourth millennium B.C. Mesopotamia. Pentagram shapes were found in excavations in Uruk (where the earliest writings were also uncovered), and in Jemdet Nasr. The ancient Sumerian city of Uruk is probably also the one mentioned in the Bible (Genesis 10) as Erech, one of the cities in the kingdom of the mighty hunter Nimrod. The pentagram was found on a clay tablet dated to about 3200 B.C. In Jemdet Nasr, pentagrams from about the same period were found on a vase and on a spindle whorl. In Sumerian the pentagram, or its cuneiform derivative, meant “the regions of the universe.” Other parts of the ancient Middle East also produced pentagrams. A pentagram on a flint scraper from the Chalcolithic period (4500-3100 B.C.) was found at Tel Esdar in the Israeli Negev Desert. Pentagrams were also found in Israel in excavations at Gezer and at Tel Zachariah, but those date to a considerably later period (the fifth century B.C.). In spite of the fact that five-pointed stars appear quite frequently in ancient Egyptian artifacts, true geometrical pentagrams are not very common, although a pentagram dating to around 3100 B.C. was found on a jar in Naqadah, near Thebes. Generally, the hieroglyphic symbol of a star enclosed in a circle meant the “underworld,” or the mythical dwelling of stars at twilight time, while stars without circles served simply to signify the night stars.
The main question we need to answer, however, in the context of this book is not whether pentagrams or pentagons had any symbolic or mystic meanings for these early civilizations but whether these civilizations were also aware of the geometrical properties of these figures and, in particular, of the Golden Ratio.
ERE BABYLON WAS DUST
Studies of cuneiform tablets dating to the second millennium B.C., which were discovered in 1936 in Susa in Iran, leave very little doubt that the Babylonians of the first dynasty knew at least an approximate formula for the area of a pentagon. The Babylonian interest in the pentagon may have originated from the simple fact that this is the figure obtained if the tips of all five fingers are pressed against a clay tablet. A line on a Susa tablet reads: “1 40, the constant of the five-sided figure.” Since the Babylonians used the sexagesimal (base 60) system, the numbers 1 40 should be interpreted at 1+40/60, or 1.666…, for the area of the pentagon. The actual area of a pentagon with a side of unit length is, in fact, not too far from this value—1.720. The Babylonians had a similar approximation for pi, the ratio of the circumference of a circle to its diameter. In fact, the approximations for both pi and the area of the pentagon relied on the same relation. The Babylonians assumed that the perimeter of any regular polygon (shape of many equal sides and angles) is equal to six times the radius of the circle circumscribing that polygon (Figure 12). This relation is actually precise for a regular hexagon (six-sided shape; Figure 12), since all the triangles have equal sides. The value of π deduced by the Babylonians was π=3⅛= 3.125. This is really not a bad approximation, given that the precise value is 3.14159…. For the pentagon, the assumption “perimeter equals six times the radius” (which is not accurate) gives the approximate value 1.666… for the area that appears in the Susa tablet.
In spite of these significant early discoveries in mathematics and the intimate relation between the pentagon-pentagram system and the Golden Ratio, there is absolutely no shred of mathematical evidence that the Babylonians knew about the Golden Ratio. Nevertheless, some texts claim that the Golden Ratio is found on Babylonian and Assyrian stelae and bas-reliefs. For example, a Babylonian stela (Figure 13) depicting priests leading an initiate to a “meeting” with the sun god is said (in Michael Schneider's entertaining book, A Beginner's Guide to Constructing the Universe) to contain “many Golden Ratio relationships.” Similarly, in an article that appeared in 1976 in the journal The Fibonacci Quarterly, art analyst Helene Hedian states that a bas-relief of an Assyrian winged demigod of the ninth century B.C. (currently in the Metropolitan Museum of Art) fits perfectly into a rectangle with dimensions that are in a Golden Ratio. Furthermore, Hedian suggests that the strong lines of the wings, legs, and beak follow other phi divisions. Hedian also makes a similar assertion about the Babylonian “Dying Lioness” from Nineveh, dated to around 600 B.C., which is currently in the British Museum in London.
Does the Golden Ratio really feature in these Mesopotamian artifacts, or is this merely a misconception?
In order to answer this question, we must be able somehow to identify criteria that will allow us to determine whether certain claims about the appearance of the Golden Ratio are true or false. Clearly, the presence of the Golden Ratio can be established unambiguously if some form of documentation indicates that artists or architects have consciously made use of it. Unfortunately, no such documentation exists for any of the Babylonian tablets and bas-reliefs.
A devoted Golden Numberist still could argue, of course, that the absence of evidence is not evidence of absence and that the measured dimensions by themselves provide sufficient proof for the employment of the Golden Ratio. As we shall soon see, however, the game of trying to find the Golden Ratio in the dimensions of objects is a misleading one. Let me illustrate this with the following simple example. Figure 14 shows a sketch of a small television set that rests on the counter in my kitchen. The drawing shows some dimensions that I have measured by myself. You will notice that the ratio of the height of the protrusion at the television's rear to its width, 10.6/6.5 = 1.63, and the ratio of the length of the front to the height of the screen, 14/8.75 = 1.6, are both in reasonable agreement with the value of the Golden Ratio, 1.618 …. Does this mean that the makers of this television decided to include the Golden Ratio in its architecture? Clearly not. This example simply demonstrates the two main short comings of claims about the presence of the Golden Ratio in architecture or in works of art, on the basis of dimensions alone: (1) they involve numerical juggling, and (2) they overlook inaccuracies in measurements.
Any time you measure the dimensions of some relatively complicated structure (e.g., a picture on a stela or a television set), you will have at your disposal an entire collection of lengths to choose from. As long as you can conveniently ignore parts of the object under consideration, if you have the patience to juggle and manipulate the numbers in various ways, you are bound to come up with some interesting numbers. Thus, in the television, I was able to “discover” some dimensions that give ratios that are close to the Golden Ratio.
The second point that is often ignored by the too-passionate Golden Ratio aficionados is that any measurements of lengths involve errors or inaccuracies. It is important to realize that any inaccuracy in length measurements leads to a yet larger inaccuracy in the calculated ratio. For example, imagine that two lengths, of 10 inches each, are measured with a precision of 1 percent. This means that the result of the measurement for each length could be anywhere between 9.9 and 10.1 inches. The ratio of these measured lengths could be as bad as 9.9/10.1 = 0.98, which represents a 2 percent inaccuracy—double that of the individual measurements. Therefore, an overzealous Golden Numberist could change two measurements by only 1 percent, thereby affecting the obtained ratio by 2 percent.
A reexamination of Figure 13 with these caveats in mind reveals, for example, that the vertical long segment has been conveniently chosen so as to include the base of the bas-relief and not just the cuneiform text. Similarly, the point to which the horizontal long segment was measured has been chosen quite arbitrarily to be to the right, rather than to the left, of the edge of the bas-relief.
Reevaluating all of the existing material in this light, I have to conclude that it is very unlikely that the Babylonians discovered the Golden Ratio.
WAY DOWN IN EGYPT LAND
The situation concerning the ancient Egyptians is more complicated, and it requires a considerable amount of detective work. Here we are confronted with what is suggested to be overwhelming evidence in the form of numerous texts that claim that phi can be found, for example, in the proportions of the Great Pyramid and in other ancient Egyptian monuments.
Let me start with two of the easier cases, those of the Osirion and the Tomb of Petosiris. The Osirion is a temple considered to be the cenotaph of King Seti I, who ruled Egypt in the XIX dynasty from about 1300 to about 1290 B.C.The temple was discovered by the noted archaeologist Sir Flinders Petrie in 1901, and the massive excavation works were completed in 1927. The temple itself is supposed to represent, via its architectural symbolism, the myth of Osiris. Osiris, the husband of Isis, was originally the king of Egypt. His brother Seth murdered him and scattered the pieces of his body. Isis collected the pieces, thus providing Osiris with a renewed life. Consequently, Osiris became king of the Underworld and of cyclic transformation through death and rebirth on both the individual and cosmic levels. After the cult of the dead was further developed during the Middle Kingdom (2000-1786 B.C.), Osiris was regarded as the judge of the soul after death.
The entire roofed Osirion temple was covered with earth, so as to resemble an underground tomb. The plan of the Osirion (Figure 15a) contains a central area with ten square columns, which is surrounded by what was probably a water-filled ditch. This structure has been interpreted to symbolize creation out of the primordial waters.
In his interesting 1982 book Sacred Geometry: Philosophy and Practice, Robert Lawlor suggests that the geometry of the Osirion is “conforming to the proportions of the Golden Section” because “the Golden Proportion is the transcendent ‘idea-form’ which must exist a priori and eternally before all the progressions which evolve in time and space.” To support his suggestion about the prominent appearance of the Golden Ratio, ö, in the architectural design of the temple, Lawlor offers detailed geometrical analyses of the type presented in Figure 15b. Furthermore, he claims that “the emphasis on the theme of the pentagon aptly symbolized the belief that the king, after death, became a star.”
In spite of their considerable visual appeal, I find Lawlor's analyses unconvincing. Not only are the lines that are supposed to indicate Golden Ratio proportions drawn at what appear to be totally arbitrary points, but even the pentagons represent, in my opinion, a rather forced interpretation of what is basically a rectangular shape. The fact that Lawlor himself presents other geometrical analyses of the temple's geometry (with ö being associated with different dimensions) further demonstrates the nonunique and somewhat capricious nature of such readings.
The situation with the Tomb of Petosiris, which was excavated by archaeologist Gustave Lefebvre during the early 1920s, is very similar. The tomb is not as old as the Osirion, dating only to about 300 B.C., and it was built for the High Priest (known as Master of the Seat) of Thoth. Since this tomb is from a period during which the Golden Ratio was already known (to the Greeks), at least in principle, the Golden Ratio could feature in the tomb's geometry. In fact, Lawlor (again in Sacred Geometry) concludes that “the Master Petosiris had a complete and extremely sophisticated knowledge of the Golden Proportion.” This conclusion is based on two geometrical analyses of a painted bas-relief from the east wall of the tomb's chapel (Figure 16a). The bas-relief shows a priest pouring a libation over the head of the mummy of the deceased.
Unfortunately, the geometrical analyses that Lawlor presents appear rather contrived (Figure 16b), with lines drawn conveniently at points that are not obvious terminals at all. Furthermore, some of the ratios obtained are too convoluted (e.g., ) to be credible. My personal feeling is, therefore, that while Lawlor's assertion that “the burial practices in the Pharaonic tradition were undertaken not merely to provide a receptacle for the physical body of the deceased, but also to make a place to retain the metaphysical knowledge which the person had mastered in his lifetime” is a very correct one, the Golden Ratio is unlikely to have been a part of Petosiris' knowledge.
I should emphasize that it is virtually impossible to prove that the Golden Ratio does not appear in some Egyptian artifacts when the evidence is presented only in the form of some measured dimensions. However, in the absence of any supporting documentation, the dimensions of the artwork or architectural design have to be such that the Golden Ratio will literally jump at you, rather than be buried so deeply to require a very complex analysis to be revealed. As we shall see later, detailed investigations of several much more recent cases for which claims existed in the literature that the artists had used the Golden Ratio show that there was no basis for these assertions.
Instead of continuing with relatively obscure objects, such as an Egyptian stela from around 2150 B.C., claimed by some to show dimensions in a ratio of ö, let me now turn to the main event—the Great Pyramid of Khufu.
PYRAMID OF NUMBERS
According to tradition, it was King Menes (or Narmer) who as a ruler of Upper Egypt conquered the rival kingdom of Lower Egypt (in the delta of the Nile), thus uniting Egypt as a single kingdom, around 3110 B.C. Sun worship as the basic form of religion was introduced under the rule of the third dynasty (ca. 2780-2680 B.C.), as were mummification and the construction of large stone monuments. The age of the great pyramids reached its climax during the fourth dynasty, around 2500 B.C., in the famous triad of pyramids at Giza (Figure 17). The “Great Pyramid” (the one at the back in the figure) stands not only as a monument to the king but also as a testimony to the success of a unified organization of the ancient Egyptian society. Researcher Kurt Mendelssohn concluded in his 1974 book The Riddle of the Pyramids that, to a large extent, the object of the whole exercise of constructing the pyramids was not the use to which the final products were to be put (to serve as tombs), but their manufacture. In other words, what mattered was not the pyramids themselves but the building of the pyramids. This would explain the apparent disparity between the tremendous effort of piling up some 20 million tons of quarried limestone and the sole purpose of burying under them three pharaohs.
In 1996 amateur Egyptologist Stuart Kirkland Wier, working under the sponsorship of the Denver Museum of Natural History, estimated that building the Great Pyramid at Giza required something like 10,000 workers. A calculation of the energy required to carry the blocks of stone from the quarry to the pyramid site, as well as that needed to lift the stones to the necessary height, gave Wier the total amount of work that had to be invested. Assuming that the construction lasted twenty-three years (the length of King Khufu's reign), and making some reasonable assumptions about the daily energy output of an Egyptian worker and about the construction schedule, Wier was able to estimate the size of the workforce.
Until recently, the dating of the pyramids at Giza relied mostly on surviving lists of kings and the lengths of their reigns. Since these lists are rare, seldom complete, and known to contain inconsistencies, chronologies generally were accurate only to within about a hundred years. (Dating by radioactive carbon contains a similar uncertainty.) In a paper that appeared in the journal Nature in November 2000, Kate Spence of Cambridge University proposed another method of dating, which gives for Khufu's Great Pyramid a date of 2480 B.C., with an uncertainty of only about five years. Spences method is the one originally suggested by the astronomer Sir John Herschel in the middle of the nineteenth century, and it is based on the fact that the pyramids were always oriented with respect to the north direction with extraordinary precision. For example, the orientation of the Great Pyramid at Giza deviates from the true north by less than 3 minutes of arc (a mere 5 percent of one degree). There is no doubt that the Egyptians used astronomical observations to determine the north direction with such accuracy.
The north celestial pole is defined as a point on the sky aligned with Earth's rotation axis, around which the stars appear to rotate. However, the axis of Earth itself is not precisely fixed; rather it wobbles very slowly like the axis of a spinning top or gyroscope. As a result of this motion, known as precession, the north celestial pole appears to trace out a large circle on the northern sky about every 26,000 years. While today the north celestial pole is marked (to within one degree) by the North Star, Polaris (known by the astronomical name of α-Ursae Mi-noris), this was not the case at the time of the Great Pyramid's construction. By tentatively identifying the two stars that the ancient Egyptians used to mark the north to be ζ-Ursae Majoris and β-Ursae Minoris, and by a careful examination of the alignments of eight pyramids, Spence was able to determine the date of accession of Khufu's pyramid to be 2480 B.C. ± 5, about seventy-four years younger than previous estimates.
Few archaeological structures have generated as much myth and controversy as has the Great Pyramid. The preoccupation with the pyramid, or the occult side of pyramidology, was, for example, a central theme to the cult of the Rosicrucians (founded in 1459 by Christian Rosenkreuz). The members of this cult made great pretensions to knowledge of the secrets of nature, magical signatures, and the like. Freemasonry originated from some factions of the Rosicrucians' cult.
The more modern interest in pyramidology started probably with the religiously permeated book of the retired English publisher John Taylor, The Great Pyramid: Why Was It Built and Who Built It? which appeared in 1859. Taylor was so convinced that the pyramid contained a variety of dimensions inspired by mathematical truths unknown to the ancient Egyptians that he concluded that its construction was the result of divine intervention. Influenced by the then-fashionable idea that the British were the descendants of the lost tribes of Israel, he proposed, for example, that the basic measuring unit of the pyramid was the same as the biblical cubit (slightly more than 25 British inches; equal to precisely 25 “pyramid inches”). This unit supposedly was also the one employed by Noah in building the Ark and by King Solomon in the construction of the Temple. Taylor went on to claim that this sacred cubit was divinely selected on the basis of the length of Earth's center-to-pole radius, with the “pyramid inch” being the five-hundred-millionth part of Earth's polar axis. His cranky book found a great admirer in Charles Piazzi Smyth, the Astronomer-Royal of Scotland, who published no fewer than three massive tomes (the first entitled Our Inheritance in the Great Pyramid) on the Great Pyramid in the 1860s. Piazzi Smyth's enthusiasm was motivated partly by his strong objection to attempts to introduce the metric system in Britain. His pseudoscientific/theological logic worked something like this: The Great Pyramid was designed in inches; the mathematical properties of the pyramid show that it was constructed by divine inspiration; therefore, the inch is a God-given unit, unlike the centimeter, which was inspired “by the wildest, most blood-thirsty and most atheistic revolution” (meaning the French Revolution). In further describing his views on the system of measures debate, Piazzi Smyth writes (in The Great Pyramid, Its Secrets and Mysteries Revealed):
So that not for the force of the sparse oratory emitted in defense of British metrology before Parliament, were the bills of the pro-French metrical agitators so often overthrown, but for the sins rather of that high-vaulting system itself; and to prevent a chosen nation, a nation preserved through history … to prevent that nation unheedingly robing itself in the accursed thing, in the very garment of the coming Anti-Christ; and Esau-like, for a little base-pottage, for a little temporarily extra mercantile profit, throwing away a birthright institution which our Abrahamic race was intended to keep, until the accomplishment of the mystery of God touching all humankind.
After reading this text, we cannot be too surprised to find out that author Leonard Cottrell chose to entitle the chapter on Charles Piazzi Smyth in his book The Mountains of Pharaoh “The Great Pyramidiot.”
Both Piazzi Smyth and Taylor essentially revived the Pythagorean obsession with the number 5 in their numerology-based analysis of the pyramid. They noted that the pyramid has, of course, five corners and five faces (counting the base); that the “sacred cubit” had about 25 (5 squared) inches (or precisely 25 “pyramid inches”); that the “pyramid inch” is the five-hundred-millionth part of Earth's axis; and so on.
Writer Martin Gardner found an amusing example that demonstrates the absurdity in Piazzi Smyth's “fiveness” analysis. In his book Fads and Fallacies in the Name of Science, Gardner writes:
If one looks up the facts about the Washington Monument in the World Almanac, he will find considerable fiveness. Its height is 555 feet and 5 inches. The base is 55 feet square, and the windows are set at 500 feet from the base. If the base is multiplied by 60 (or five times the number of months in a year) it gives 3,300, which is the exact weight of the capstone in pounds. Also, the word “Washington” has exactly ten letters (two times five). And if the weight of the capstone is multiplied by the base, the result is 181,500—a fairly close approximation to the speed of light in miles per second.
Here, however, comes the most dramatic announcement concerning the Great Pyramid in the context of our interest in the Golden Ratio. In the same book, Gardner refers to a statement that, if true, shows that the Golden Ratio was actually incorporated in the Great Pyramid by design. Gardner writes: “Herodotus states that the Pyramid was built so the area of each face would equal the area of a square whose side is equal to the Pyramid's height.” The Greek historian Herodotus (ca. 485-425 B.C.) was called “the Father of History” by the great Roman orator Cicero (106-43 B.C.). While Gardner did not realize the full implications of Herodotus' statement, he was neither the first nor the last to present it.
In an article entitled “British Modular Standard of Length,” which appeared in The Athenaeum on April 28, 1860, the famous British astronomer Sir John (Frederick William) Herschel (1792-1871) writes:
The same slope,… belongs to a pyramid characterized by the property of having each of its faces equal to the square described upon its height. This is the characteristic relation which, Herodotus distinctly tells us, it was the intention of its builders that it should embody, and which we now know that it did embody.
Most recently, in 1999, French author and telecommunications expert Midhat J. Gazalé writes in his interesting book Gnomon: From Pharaohs to Fractals: “It was reported that the Greek historian Herodotus learned from the Egyptian priests that the square of the Great Pyramid's height is equal to the area of its triangular lateral side.” Why is this statement so crucial? For the simple reason that it is equivalent to saying that the Great Pyramid was designed so that the ratio of the height of its triangular face to half the side of the base is equal to the Golden Ratio!
Examine for a moment the sketch of the pyramid in Figure 18, in which a is half the side of the base, s is the height of the triangular face, and h is the pyramid's height. If the statement attributed to Herodotus is correct, this would mean that b2 (the square of the pyramids height) is equal to s × a (the area of the triangular face; see Appendix 3). Some elementary geometry shows that this equality means that the ratio s/a is precisely equal to the Golden Ratio. (The proof is given in Appendix 3.) The immediate question that comes to mind is: Well, is it? The base of the Great Pyramid is actually not a perfect square, as the lengths of the sides vary from 755.43 feet to 756.08 feet. The average of the lengths is 2a = 755.79 feet. The height of the pyramid is h = 481.4 feet. From these values we find (by using the Pythagorean theorem) that the height of the triangular side s is equal to s = 612.01 feet. We therefore find that s/a = 612.01/377.90 = 1.62, which is indeed extremely close to (differing by less than 0.1 percent from) the Golden Ratio.
Taken at face value, therefore, this evidence would imply that the ancient Egyptians indeed knew about the Golden Ratio, since not only does this number appear in the ratio of dimensions of the Great Pyramid but its presence seems to be supported by historical documentation of the intentions of the designers, in the form of Herodotus' statement. But is this true? Or are we witnessing here what the Canadian mathematician and author Roger Herz-Fischler called “one of the most ingenious sleights of hand in ‘scientific’ history”?
Clearly, since the measurements of the dimensions cannot be altered, the only part in this “evidence” for the presence of the Golden Ratio that can be challenged is Herodotus' statement. In spite of the numerous repetitions of the quote from History, and even though one cannot cross-examine a man who lived 2,500 years ago, at least four researchers have taken upon themselves the “detective” work of investigating what Herodotus really said or meant. The results of two of these inquiries have been summarized by Herz-Fischler and by University of Maine mathematician George Markowsky.
The original text from Herodotus' History appears in paragraph 124 of book II, named Euterpe. Traditional translations read: “Its base is square, each side is eight plethra long and its height the same,” or “It is a square, eight hundred feet each way, and the height the same.” Note that one plethron was 100 Greek feet (approximately 101 English feet).
These texts look very different from what has been presented as a quote (that the square of the height equals the area of the face) from Herodotus. Furthermore, the figures for the pyramid's dimensions that Herodotus mentions are wildly off. The Great Pyramid is far from being 800 feet high (its height is only about 481 feet), and even the side of its square base (about 756 feet) is significantly less than 800 feet. So where did that “quote” come from? The first clue comes from Sir John Herschel's article in The Athenaeum. According to Herschel, it was John Taylor in his book The Great Pyramid, Why Was It Built and Who Built It? who had “the merit of pointing out” this property of the pyramid and Herodotus' quote. Herz-Fischler tracked down the misconception to what appears to be nothing more than a misinterpretation of Herodotus in John Taylor's by now infamous book.
Taylor starts with a translation of Herodotus that does not read too differently from the ones above: “of this Pyramid, which is four-sided, each face is, on every side 8 plethra, and the height equal.” Here, however, he lets his imagination run wild, by assuming that Herodotus meant that the number of square feet in each face is equal to the number of square feet in a square with a side equal to the pyramid's height. Even with this “imaginative” interpretation, Taylor is still left with the small problem that the number mentioned (eight plethra) is way off the actual measurements. His suggested solution to this problem is even more appalling. With no justification whatsoever, he claims that the eight plethra must be multiplied by the area of the base of one of the small pyramids standing on the east side of the Great Pyramid.
The conclusion from all of this is that Herodotus' text can hardly be taken as documenting the presence of the Golden Ratio in the Great Pyramid. The totally unfounded interpretation of the text instigated by Taylor's book (and subsequently repeated endless times) really makes little sense and represents just another case of number juggling.
Not everyone agrees with this conclusion. In an article entitled “The Icosahedral Design of the Great Pyramid,” which appeared in 1992, Hugo F. Verheyen proposes that the Golden Ratio as a mystic symbol may have been deliberately hidden within the design of the Great Pyramid “as a message for those who understand.” As we shall see later, however, there are more reasons to doubt the idea that the Golden Ratio featured at all in the pyramid's design.
When we realize that the Great Pyramid rivals the legendary city of Atlantis in the numbers of books written about it, we should not be too surprised to hear that ö was not the only special number to be invoked in pyramidology—π was too.
The π theory appeared first in 1838, in Letter from Alexandria, on the Evidence of the Practical Application of the Quadrature of the Circle, in the Configuration of the Great Pyramids of Egypt, by H. Agnew, but it is generally credited to Taylor, who merely repeated Agnew's theory. The claim is that that ratio of the circumference of the base (8a in our previous notation, in which a was half the side of the base) to the pyramid's height (h) is equal to 2π. If we use the same measured dimensions we used before, we find 8a/h = 4 × 755.79/481.4 = 6.28, which is equal to 2π with a remarkable precision (differing only by about 0.05 of a percent).
The first thing to note, therefore, is that just from the dimensions of the Great Pyramid alone, it would be impossible to determine whether phi or pi, if either, was a factor in the pyramid's design. In fact, in an article published in 1968 in the journal The Fibonacci Quarterly, Colonel R. S. Beard of Berkeley, California, concluded that: “So roll the dice and choose your own theory.”
If we have to choose between π and ö as potential contributors to the pyramid's architecture, then π has a clear advantage over ö. First, the Rhind (Ahmes) Papyrus, one of our main sources of knowledge of Egyptian mathematics, informs us that the ancient Egyptians of the seventeenth century B.C. knew at least an approximate value of π, while there is absolutely no evidence that they knew about ö. Recall that Ahmes copied this mathematical handbook at about 1650 B.C., during the period of the Hyksos or shepherd kings. However, he references the original document from the time of King Ammenemes III of the Twelfth Dynasty; and it is perhaps not impossible (although it is unlikely) that the contents of the document had already been known at the time of the construction of the Great Pyramid. The papyrus contains eighty-seven mathematical problems preceded by a table of fractions. There is considerable evidence (in the form of other papyri and records) that the table continued to serve as a reference for nearly two thousand years. In his introduction, Ahmes describes the document as “the entrance into the knowledge of all existing things and all obscure secrets.” The Egyptian estimate of πappears in problem number 50 of the Rhind Papyrus, which deals with determining the area of a circular field. Ahmes' solution suggests: “take away of the diameter and square the remainder.” From this we deduce that the Egyptians approximated π to be equal to = 3.16049…, which is less than 1 percent off the correct value of 3.14159….
A second fact that gives πan advantage over ö is the interesting theory that the builders incorporated πinto the pyramid's design even without knowing its value. This theory was put forward by Kurt Mendelssohn in The Riddle of the Pyramids. Mendelssohn's logic works as follows. Since there is absolutely no evidence that the Egyptians at the time of the Old Kingdom had anything but the most rudimentary command of mathematics, the presence of πin the pyramid's geometry must be the consequence of some practical, rather than theoretical, design concept. Mendelssohn suggests that the ancient Egyptians may have not used the same unit of length to measure vertical and horizontal distances. Rather, they could have used palm fiber ropes to measure the height of the pyramid (in units of cubits) and roller drums (one cubit in diameter) to measure the length of the base of the pyramid. In this way, horizontal lengths would have been obtained by counting the revolutions in units one might call “rolled cubits.” All the Egyptian architect then had to do was to choose how many cubits he wanted his workers to build upward for every horizontal rolled cubit. Since one rolled cubit is really equal to π cubits (the circumference of a circle with a diameter of one cubit), this method of construction would imprint the value of π into the pyramid's design without the builders even knowing it.
Of course, there is no way to test Mendelssohn's speculation directly. However, some Egyptologists claim that there does exist direct evidence suggesting that neither the Golden Ratio nor pi were used in the Great Pyramid's design (not even inadvertently). This theory is based on the concept of the seked. The seked was simply a measure of the slope of the sides of a pyramid or, more precisely, the number of horizontal cubits needed to move for each vertical cubit. Clearly, this was an important practical concept for the builders, who needed to keep a constant shape with each subsequent block of stone. The problems numbered 56 to 60 in the Rhind Papyrus deal with calculations of the seked and are described in great detail in Richard J. Gillings's excellent book, Mathematics in the Time of the Pharaohs. In 1883, Sir Flinders Petrie found that the choice of a particular seked (slope of the pyramid's side) gives for the Great Pyramid the property of “ratio of circumference of the base to the pyramid's height equal to 2π” to a high precision, with π playing no role whatsoever in the design. Supporters of the seked hypothesis point out that precisely the same seked is found in the step pyramid at Meidum, which was built just before the Great Pyramid at Giza.
Not all agree with the seked theory. Kurt Mendelssohn writes: “A great number of mathematical explanations have been suggested and even one, made by a noted archaeologist [Petrie], that the builders by accident used a ratio of , which is very close to 4/π], remains lamentably unconvincing.” On the other hand, Roger Herz-Fischler, who examined no fewer than nine theories that have been advanced for the Great Pyramid's design, concluded in a paper that appeared in the journal Crux Mathematicorum in 1978 that the seked theory is very probably the correct one.
From our perspective, however, if either of the two hypotheses, seked or rollers, is correct, then the Golden Ratio played no role in the Great Pyramid's design.
Is, therefore, the 4,500-year-old case of the Golden Ratio and the Great Pyramid closed? We would certainly hope so, but unfortunately history has shown that the mystical appeal of the pyramids and Golden Numberism may be stronger than any solid evidence. The arguments presented by Petrie, Gillings, Mendelssohn, and Herz-Fischler have been available for decades, yet this has not prevented the publication of numerous new books repeating the Golden Ratio fallacy.
For our purposes, we have to conclude that it is highly unlikely that either the ancient Babylonians or the ancient Egyptians discovered the Golden Ratio and its properties; this task was left for the Greek mathematicians.