Pyramid Quest (43 page)

Phi (or the Golden Number, Φ) is equal to (1 + square root of 5) / 2 = approximately 1.6180339 . . . (see West, 1979, pp. 74, 75).

Phi is obtained by dividing a line, AC, at a point B such that AC / AB = AB / BC. That is, the whole to the longer part is the same proportion as the longer part to the shorter part, both of which equal Φ. This is the Golden Section, or Primordial Scission (West, 1979, p. 74, discussing the work of Schwaller de Lubicz).

Take a square with a side of unit 1 and cut it in half from one side to the other to form two rectangles of 1 × ½. The diagonal of one of the rectangles plus ½ equals Φ. By the Pythagorean theorem, the length of the diagonal under consideration, call it
W,
bears the following relationship to the other two sides:
W
²
= 1
²
+ (1/2)
²
. Or,
W
²
= 1.25 and thus
W
= square root of 1.25, and Φ = square root of 1.25 + (1/2). However, the square root of 1.25 can be multiplied by 1 in the form of √4 /2 to arrive at √4x1.25 / 2 = √5 / 2. Now substitute √5/ 2 for √1.25 in the equation Φ = √1.25 + (1/2), and we arrive at Φ = (1 + √5)/ 2.

One of the important properties of Φ is that: 1 + Φ = Φ
²

In the Fibonacci sequence—1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .—each number is the sum of the two preceding numbers. Ratios of successive terms give increasingly better approximations of phi (G, Golden number, Φ): for example, 55 / 34 = 1.61747, whereas phi is an irrational number 1.6180 . . . incapable of being expressed as a quotient of integers (see Herz-Fischler, 2000, p. 242; Tompkins, 1971, p. 192). It is through the Fibonacci series that phi is said to control many things in nature, such as growth curves of marine organisms (such as the outwardly spiraling shell of the Nautilus), seed whorls in composite flowers, or the structure of a spiral galaxy, and so on.

According to Schwaller de Lubicz (see Tompkins, 1971, p. 194), the ancient Egyptians had figured out that the relationship between π and Φ is π = Φ
²
x 6 / 5. Take two approximations of Φ in the Fibonacci series in sequence and substitute them into this equation, and you can produce a good approximation of π (the π approximations get better as one goes further along in the Fibonacci series). As an example, an approximation of π apparently used in the Great Pyramid is:

Stecchini, a modern researcher, has argued that the Great Pyramid was designed with phi in mind, at least in part. Let
Y
be the horizontal length from the middle of the northern side at the base to a point directly under the apex of the Great Pyramid;
Y
equals ½ of the standard base length of 439½ cubits (according to Stecchini; see hereafter) divided by 2 (230.363178 meters divided by 2 equals 115.181589 meters), according to Stecchini (1971, p. 368). To say that the northern face was designed with phi in mind means that
Y
divided by the square root of 1 over phi equals the height of the Great Pyramid, or 115.181589 meters / √1/1.618) = 146.512 meters. This corresponds to what Herz-Fischler (2000, pp. 80-91) describes as the “Kepler Triangle Theory” of the shape of the Great Pyramid. If
A
is the apothem of a side of the Great Pyramid (the apothem is the length from the middle of the base of one side of the pyramid to the apex or summit of the pyramid; in the Great Pyramid the apothem would be approximately 186.5 meters if the summit came to a point; if the sides are not exactly the same, then each may have its own apothem value), then, according to the Kepler triangle theory,
A
/
Y
= Φ.

The equivalence of these two approaches can be shown as follows.

By the Pythagorean theorem:
Y
²
+
h
²
=
A
²
. Substitute
Y
/ √1/Φ for
h
in the last equation, and one has

However, one of the properties of Φ is that (1 + Φ) = Φ
²
(Herz-Fischler, 2000, p. 83), so Φ
²
Y
²
=
A
²
, or Φ
Y
=
A,
and rearranging,
A
/
Y
= Φ.

A Kepler triangle is a right triangle where the ratio of the hypotenuse to the larger side equals the ratio of the larger side to the smaller side; in a Kepler triangle, the hypotenuse divided by the smaller side equals Φ (Herz-Fischler, 2000, pp. 81-82). In the foregoing equation
A
/
Y
= Φ,
A
is the hypotenuse, and
Y
is the smaller adjacent side. In the case of the Great Pyramid, if we use the following values for the apothem, height, and
Y,
respectively: 186.367 meters (apothem value calculated from the next two values using the Pythagorean theorem), 146.512 meters, 115.182 meters; then the ratio of the hypotenuse to the larger side is 1.270, and the ratio of the larger side to the smaller side is 1.272, which can be considered a fairly close match.

Related to the Kepler triangle theory, in that it gives ultimately the same result, namely that
A
/
Y
= Φ, is the so-called equal area theory (Herz-Fischler, 2000, pp. 96-111). The core of the equal area theory is that the surface area of one face of the Great Pyramid is equal to the area of the square of the height. In terms of
h, A,
and
Y
used earlier, the equal area theory states that:

Using the Pythagorean theorem, we know that h
²
+
Y
²
=
A
²
.

Rearranging,
h
²
=
A
²
-
Y
²
, and substituting into the equation
h
²
=
AY,
we have:

Divide both sides by
Y
²
, and we have (
A
/
Y
)
²
- 1 =
A
/
Y
, then add 1 to each side, and we have 1 +
A
/
Y
= (
A
/
Y
)
²
, remembering that 1 + Φ = Φ
²
.

This means that
A
/
Y
= Φ, which is the same ultimate result as the Kepler triangle theory.

If
A
/
Y
= Φ, then 1 / Φ =
Y
/
A,
and by trigonometry, the theoretical slope of a side of the Great Pyramid is equal to the inverse cosine of 1 / Φ = 1 / 1.6180 = 0.6180, which is approximately 51.827°.

Note that in terms of the slope angle each predicts, the pi theory can be considered closer to the actual observations of the slope of the Great Pyramid than the Kepler triangle theory or equal area theory. All three, however, really give remarkably close results to the actual observations (which observations may include some degree of inaccuracy from the slopes and angles possibly intended by the original architects).

The equal area theory was also espoused by Taylor (1859), and was at least partially suggested, but not developed, by Agnew (1838, as quoted by Herz-Fischler, 2000, p. 98). Herz-Fischler thinks it is “not unlikely” (p. 99) that Taylor was inspired by Agnew’s comments. If anyone deserves credit for fully developing the equal area theory, I think it is Taylor.

Agnew and Taylor both based their concepts (or, in Agnew’s case, protoconcept) of the equal area theory on a reinterpretation of Herodotus. Herz-Fischler (2000, p. 98) quotes a translation of the relevant passage from Herodotus’s
Histories,
book 2, chapter 124, as follows: “The pyramid itself was twenty years in the making. Its base is square, each side is eight plethra long and its height is the same; the whole is of stone polished and most exactly fitted; there is no block of less than thirty feet in length.”

Taken literally in terms of linear dimensions, Herodotus’s statement cannot be correct. The length of the sides of the Great Pyramid are not equal to its height, and the lengths of the sides are not even equal to their apothems or the arris of the Great Pyramid (the arris is the length of the edge between two sides, from the corner of the pyramid to the vertex, about 219 meters in the Great Pyramid). Taylor suggested that the term
plethron
(plural
plethra
) was being used here by Herodotus as a measure of surface area, not as a linear measure, and indeed it could be used in either sense in antiquity (as
plethra
was used elsewhere by Herodotus as a measure of surface area). It is relatively easy to see how the surface area of each side can be measured in terms of surface area, but what does it mean to measure the height in terms of surface area? Taylor suggested that what Herodotus means is that the square of the height (a surface area that is
h
×
h
) must be equal to the surface area of one side.

In terms of this interpretation, it is unclear to me exactly what a plethron was in terms of modern units. Herz-Fischler (2000, p. 100) equates 8 plethra with 7,589 square meters, but I am not convinced this is correct. What is important for the equal area theory, however, is the closeness of the fit of the area of a side to
h
²
. If we use a value of 146.6 meters for
h,
then
h
²
= 21,492 square meters. Using 230.4 meters for the length of one side, and 186.5 meters for the length of an apothem, the surface area of a side is 21,485 square meters. (The values used here for height, side length, and apothem are those used by Herz-Fischler in his book.) This is a discrepancy of only 7 square meters, so the theory and the calculations are very close.

Stecchini (1971, pp. 370-372) traces other ancient accounts of the dimensions of the Great Pyramid back to Agatharchides of Cnidus, of the late second century B.C., who served under the Ptolemaic kings of Egypt. According to Stecchini’s interpretation of these writings, the “surface area interpretation” of Herodotus is correct.

Smyth, in the first edition
Our Inheritance in the Great Pyramid
(1864) never mentioned the equal area theory, despite being the foremost champion of Taylor’s work on the Great Pyramid. Smyth only mentions the equal area theory in footnotes in later editions. Instead, Smyth promoted and supported the pi theory. Robert Ballard (1882) concluded that
Y
/
A
(apothem to half the length of a side) is 34 / 21, which is a close approximation to Φ, and he used this to support a version of the equal area theory. Martin Gardner (1957, p. 178), the well-known debunker, apparently accepted the equal area theory for the Great Pyramid, saying:

Most Great Pyramid researchers assume that initially the structure was at least intended to have a perfectly square base and sides that rose to the apex at equal angles. Stecchini (1971), on the basis of his analysis, questions these fundamental assumptions. Stecchini (1971, p. 368) believes 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 this was then modified in the final plan and construction. According to Stecchini (1971, p. 364), the basic length of each side was modified to 439.5 cubits (230.363 meters, using Stecchini’s value of 524.1483 millimeters for the cubit used in the Great Pyramid). The perimeter of the Great Pyramid was therefore meant to be 1,758 cubits (921.453 meters), which, according to Stecchini (1971, p. 365) is half the value of a minute of latitude at the equator, as calculated by the ancient Egyptians as 3,516 cubits (1,842.905 meters; Stecchini, 1971, p. 365, cites 1,842.925 as a modern calculation of the same entity).

Stecchini (1971, p. 364) interprets data from the Cole Survey (1925) to indicate that the sides of the Great Pyramid were not off from a perfect square due to accidents or inaccuracies in the layout but that the base was purposefully designed to be slightly different from a perfect square. Stecchini believes the western side was laid out (the line for its alignment drawn) first, and the northern side was made to be as perfectly perpendicular to it as possible. The eastern side, however, was intended to be an angle of 3’ greater than perpendicular to the northern side (i.e., the northeast corner was meant to be 90° 03’ 00”), and the southern side was intended to be ½‘ greater than perpendicular (i.e., the southwest corner was meant to be 90° 00’ 30”; see Stecchini, 1971, pp. 364-365). Furthermore, using the data from the Cole Survey on a small line found on the pavement at the base of the Great Pyramid near the middle on the north side, which some have assumed to be the original axis of the Great Pyramid, Stecchini (1971, pp. 366-367) concludes that the north-south axis of the pyramid was off center, and accordingly the apex was off center, by about 35.5 millimeters to the west (the line in question is located 115.090 meters from the northwest corner and 115.161 meters from the northeast corner). The foregoing information would indicate that the four faces of the Great Pyramid have slightly different slopes from each other, as was suspected but not actively pursued by Petrie (1883, 1885).