Mark Reynolds667 Miller Avenue Mill Valley, CA 94941 USA
In the measurement and geometric analysis of an ancient structure, it is often necessary to combine scientific and mathematical objectivity with an open mind that will permit an understanding of what appears to be the intentions of the maker…that is to say, educated common sense. Ideally, one would rejoice in having the original plans and documents that would clearly explain the precise measures, or to have an opportunity to measure the work as it was at its completion. More often, the problem can be reasonably solved by researching documents and texts on the piece to be analyzed, and by looking for reputable and reliable scholars that have gone before to do similar studies. By combining this approach with drawings and photographs and, where possible, with measurements in situ, a sensible study can be undertaken Scholars and writers have presented studies, surveys and mathematical information to support their beliefs that the two pyramids under discussion incorporate mathematical formulae that refer to pi, the size of the earth, the 'squaring of the circle' and other numerical and esoteric data. Some of the information is valid, some conjecture, while other items in print must be viewed at least skeptically, if not completely ignored. Of course, there is always the difficulty presented by the mathematical certitude that is perpetually in conflict with the intuitive nature of the artist. Mathematicians and artists will disagree on whether a line has thickness, and scholars will argue over the ability of a geometer to rectify the square or to be completely accurate with an irrational number.[2]Such are the perils of anyone stepping into the arena of pi, phi, theta, the root
rectangle system, and other incommensurables. At the risk of
incurring arguments over these issues,
let us continue, with the hope that the mathematician and the
artist will reach a common ground of mutual respect and agreement
in their shared appreciation for the beauty involved.
pi to determine if one was more
accurate than the other. I was quite content to use rational
approximations for pi [3],
such as the Egyptian value of
256/81 or the 22/7 of Archimedes of Syracuse, one of my heroes.
(He was murdered by Roman Soldiers for doing geometric constructions
in the dirt.) I find this approach more helpful in appreciating
the flavor of the works, and the intentions of the master masons
who were involved in constructing them. However, a calculator
is indispensable for calculating percentatage deviations. Below is a chart that will be used in the presentation of the comparative analyses of the two pyramids. It contains the measures I used in my study of the two pyramids.[4]
The rectangles generated from the height of each pyramid and its ½ base, when the ½ base is considered to be 1:
The ratio of the base perimeter to the height:
The ratios of the bases to one another:
As stated by Graham Hancock and others, the two pyramids are "almost" or "very nearly equal" to one another in base perimeter. (Along with this thought there is also the statement that the Pyramid of the Sun is "almost" half the height of the Great Pyramid; this will be addressed later.) Because of the closeness, there is often the tendency to state that they are equal. One of the major points of the paper is that there is a difference, an ever-so-slight one that can be explained and demonstrated by the geometric constructions presented here. The Great Pyramid is 1.03... times larger than the base of the Pyramid of the Sun. Conversely, the base of the Pyramid of the Sun is 97…% of the Great Pyramid's base. Is this an accidental similarity, or a telling coincidence? Were the two monuments built by totally separate cultures, or was there a shared cultural code?As a geometer, my curiosity was piqued by these questions and issues, and I wanted to proceed to geometric constructions to investigate the relationships, especially because the structures under discussion were made by cultures who revered geometry as a gift from the gods. Our brethren of ancient architecture probably did not have advanced calculating technologies, and it may be possible that they were not so concerned with incommensurable, irrational, and transcendental numbers, for in the geometry, these qualities exist by construction, not measurement. Measure was used primarily for calculation of heavenly and earthly phenomena. However, they were keenly aware of circles and cycles and spheres, for these things were visually observed. What is evident now is the fact that they were concerned about relationships surrounding the concepts of unity, duality, and multiplicity, of how all could be melded into ONE. Schwaller de Lubicz writes in The Temple of Man that, for the Egyptians,
ONE is the largest number, and all other numbers are fractions
of this primal and universal unity.[5]
The real issue isn't how accurately the priestly geometers could calculate and use pi and phi, but rather, how they could integrate
the two into one cohesive unity. The greatest difficulty with
this coupling is that of the "square peg in a round hole",
or in geometer's terms, the classical problem of Quadrature,
Squaring the Circle. There are significant relationships between
the circle and the square, but there is a fundamental difference
between them, not only geometrically or numerically, but symbolically
and philosophically as well. The circles and spheres in the dome
of the sky are contrasted by the four directions of the Earth
and its four seasons. It is difficult to find squares above and
circles below. The ancient master builders sought the marriage,
the union, of these opposites in the temple; to them, this was
the extension of spirit into matter.Through all of this, the thread that bound it all together was the search for the geometry that tied pi and phi
together. This union can be consummated with esoteric
relationships:
- 4 divided by root-
*phi*is approximately equal to*pi*(4/1.2720196 =3.1446055, a percentage deviation of less than 0.01%). - 5/6 of
*phi*squared is approximately equal to*pi*(6/5 x 2.618034= 3.1416408, a percentage deviation of less than 0.002%). *phi*-squared plus 1, divided by*phi*is approximately equal to 2 ([1.618034 + 1]/1.618034 = 2.236068, a percentage deviation of 11.8%).
Arguably, there is no true equality in these figures because
we are dealing with roots, irrationals, and transcendentals.
The importance lay in the attempt to make their creations as
perfect as possible with what was there. The question is only
one of localities or global communities: if the Egyptians figured
this way, did their counterparts on the other side of the world
independently and coincidentally figure things in the same way?
Or, were the cultures building pyramids a shared group of masons
traveling the globe, doing similar things in greatly differing
places? There may not be a definitive answer, but I suggest that
there is a unique and specific geometric construction that uses
the circle and the square, and
G is a golden section of
Rectangle : AR
:: 1 : phi. Now that the golden section rectangle has been generated, we will now proceed to draw a new rectangle called the square root of the golden section (Figure 4).
The triangle that is formed,
,
we can begin to construct the geometry of the pyramids. In
this section, I will be giving procedures and explanations for
the geometric constructions found in the paper. Rather than presenting
cold and technical procedures, I would like to briefly make a
few points on the philosophical aspects of the geometer's art.
With these thoughts in mind, it is my hope that the constructions
will take on added meaning. The geometer believes that as the drawing is done, the geometry exerts a certain quality to its maker, similar to the gold in the alchemist's cucurbit. Geometric devices are powerful in and of themselves as well as for the purposes for which they are drawn. The geometer will usually begin a drawing from the circle, square, and/or the triangle, with the square being the most frequently used, and the most practical. As a general rule, philosophical geometry considers pi to be in the realm of the circle,
phi in that of the square, roots and rectangles in that
of the square as well, and polygons and regular solids in that
of the circle/sphere and the square/cube. The most important points to keep in mind when examining the drawings presented below are: - All the drawings here emanate from and are generated by the square.
- The side of this square is always ONE (=Unity or wholeness); understanding this, and that all the other numbers that are discussed are in relation to this ONE, is the simplest way to comprehend the other measures, as they all rely on 1 as the basis for comparing their magnitudes and ratios.
- The square is the expression of Unity on the Earth, and that the circle is the same expression of Unity for that which is above and around.
- All rectangles are the expansion of the square.
- The orientation of the rectangle emphasizes either the horizon of the earth, or the vertical relationship between that which is below with that which is above.
- The geometric construction attempts to manifest a higher absolute system into the physical world; as such, it attempts to represent both rational and irrational magnitudes; in philosophical, or sacred, geometry, there are absolute figures (circle, square, triangle, cross, spiral) in a higher realm that become relative when manifested in the physical realm (variations occur, for example, when various line weights and media are used).
- In spite of the perfection of what I call "Ideal Numbers"
(e.g., (root-5 + 1)/2 =
*phi*, or 1/*phi*), coupled with the precision of the geometer's compasses, straightedge, and set squares, it is impossible to make a perfect drawing or building; thus the eternal duality of the absolute vs. the relative.
It is also important to note that, in the procedures for the
geometric constructions used in the Pyramids, there is an underlying
organic quality that functions in the drawings. First, the half-diagonal
of the square (root-5 / 2 ) is moved to generate the golden section.
This is done by attaching the magnitude 1.118033…to one-half
(.5) the side of the square (0.5 + 1. 11803…= 1.618033…).[8] Second,
a very specific organic action is taken on a function that is
in and of itself already organic. This step requires that the
golden section length be moved from its upright 90-degree angle
to become the root of the new rectangle called the root- the
nature of geometry itself to manifest growth, and phi
is its most organic function.Beginning with the Triangle of Price AQZ in Figure 4 and using bilateral
symmetry, we can construct the Great Pyramid's elevation by doubling
the base AZ at Z to be AL (do this by placing
the compasses on Z to swing AZ to AL). Next,
draw the slope QL to complete the elevation (Figure
5). The resulting triangle, AQL,
is now an isosceles triangle with sides = 1.618, 2, and 1.618,
and the vertical axis (called cathetus) is = 1.2720196…,
or root-phi. (Points p3 and M in this figure
will be discussed later.)Figure 6 combines the triangular elevation of the Great Pyramid ( AQN) with the plan of the square
base (BKJL), seen together in one drawing. This is an
important drawing because it plays a key role in later drawings
of both pyramids, enabling us to see how both have been formed.We will now construct the slightly smaller
base of the Pyramid of the Sun. Starting with the drawing of
the the combined elevation and plan of the Great Pyramid shown
in Figure 6, use the center Q1 to inscribe a circle inside
the square of the base and tangent to the four sides at the midpoint
p of the square. (In philosophical geometry, the highest
shape in the hierarchy is the circle, for it represents the cosmos.
On Earth, this hierarchy begins with the square. Constructing
the circle in the square is one form of mandala, and can be viewed
as the circle being born of the square.) The circle cuts the
slopes AQ and QN of the Great Pyramid elevation
at the points p3. It can be seen that these two points
are not at the same level and height as M. The difference
is subtle, almost infinitesimal, but it is there. These two points
will work in tandem and harmony with the next drawing.In Figure 7, point p3 are used to establish point p4, which
is at the same height as the point p3. Point p4 will
mark the radius Q1p4 of a new, smaller, concentric circle
just within the first circle. This new circle will yield the
size of the base of the Pyramid of the Sun, the base that is
"almost", or, "not quite", equal to the base
of the Great Pyramid.
Now we come to the more difficult drawings, but difficult
tasks bring greater rewards. The goal of the constructions is
to obtain the height, the elevation of the second Pyramid. Ascertaining
the results is arduous because of the destruction and abuse the
structure has suffered over the centuries, most especially in
the twentieth century. Unlike the Great Pyramid, which has a
basically straight slope and a clearly marked capstoneapex--albeit
with a 9 meter high pole placed on the degraded top of the pyramid--the
Sun Pyramid has courses almost like terraced steps, and its true
top, possibly containing an altar space or temple, no longer
exists. These slopes are at varying angles for various reasons.[9] - Rectangle
*AKMZ*is a root-*phi*rectangle. - Rectangle
*APRZ*is also a root-*phi*rectangle. Here, it is considered to be a reciprocal rectangle, 1/root-*phi*, as it has been constructed from the diagonal*KZ*. *AR*is a reciprocal/diagonal. It meets the requirement of the reciprocal relationship with the diagonal of the (any) rectangle*KZ*in that it crosses the diagonal at 90 degrees. This is true at any point along the diagonal*KZ*.- When this point is at
*OC*, the result of the reciprocal being drawn from one of the corners, here point*A*,*OC*is called an "occult center", as it is "hidden from the eye". All rectangles (except the square) have four occult centers, as there are four corners to the rectangle. These four occult centers frame the dead center of all rectangles. - When the reciprocal is drawn as it is here, we have a diagonal/reciprocal
relationship that generates a geometric progression that is based
on the ratio of the rectangle in which it is drawn. Here, that
ratio of the rectangle is 1
**:**1. 2720196…, so the four lengths*OCR*,*OCZ*,*OCA*,*OCK*= 1, 1.272…, 1. 618…, 2.058… - ONLY In the root-
*phi*rectangle, the occult centers are at the golden sections of the heights and widths of the rectangle. - Where the diagonal
*KZ*cuts*PR*at*q*is also an occult center!
Figure 9
shows the division of the area into the reciprocal, and how it
generates point *PKrq*=*qrMR*= root-*phi**PKrq*is the reciprocal rectangle to the master rectangle*qrMR*.- Point
*r*is at a golden section of*KM* - Point
*q*is at a golden section of*PR*.
These construction points are esoteric in the truest sense
of the word, and fall outside the realm of "normal"
geometric analysis. They are not used or found in the usual procedures
of construction. A rather complex grid has to be developed, and
extra steps are needed to generate the specific eyes [11]
used in the field of the gridwork.
I do, however, believe that the masons of the Pyramids knew of
them, and of the unique qualities that I've outlined here. Only
someone who worked or works with the ratio and the grid work
for a long time would know where these "eyes" are located;
only someone or some group that was incredibly close to the drawing,
as I have been, would use the points that I am discussing here.
These points bear no cosmic or sacred meaning for the novice,
hence the esoteric nature of the constructions. *LR2***:***R2Z***::***R2Z***:***ZA*- 1/
*phi***:**1**::**1**:***phi* *LZ*=*ZA*= 2
Triangle
Point
Lines *GOZ*=*ZON**GON*is the elevation of the Pyramid of the Sun- If a rectangle be made with short side
*ZO,*long side*GZ*as the long side, and diagonal*GO*, the ratio of each of the two rectangles yields a ratio of*GZ***:***OZ*= 1**:**0 .6369339…, roughly 1/*phi*.[13]
Had the apex gone to the exact half way point (designated
by the base measure
phi, not phi.
I have done a great number of drawings and studies since then
on the ratio, both in the rectangle and the triangle, as well
as a good deal of work on the elevation of the Great Pyramid.
This has brought me to a deep understanding of the number and
the ratio. I believe it to be one of the basic building blocks
of the natural world, and I have come to have great regard for
the geometric relationships it contains and creates. It is not
surprising to me that the Egyptian masters encoded this relationship
into the pyramid, nor indeed that it was used for the Pyramid
of the Sun.There is also an astonishing relationship that I will mention now. Figure 19 shows the relative sizes of the Earth and our Moon. The square root of the golden section, 1.2720196… and the Great Pyramid, constructed by this number, both yield the sizes of these two heavenly orbs. This specific technique is one of the keys to understanding how the two pyramids are linked, for it is precisely where the arc from this rotation cuts the sides of the triangle AQL that yields the difference
in the dimensions of the two bases we say in Figure 7.The moon is about 27.3% the size of
the earth, or, 1.272…minus 1. Figure 20 shows this relationship,
as well as the squaring of the circle. Although the problem of
squaring the circle was proven mathematically impossible in the
19th century (as pi, being irrational, cannot be exactly
measured), the Earth and the moon, the Great Pyramid and the
square root of phi are related to it, all coming about
as close as you can get to the solution!In Figure 20: - The half-base of the Great Pyramid
*AZ*is ONE (Unity), and is equal to the radius of the Earth. - The base of the Great Pyramid
*AL*is equal to 2. *L*=*GN*=*GK*=*KB*=*BN*= 2; therefore*GKBL*is a square.- The circle inscribed within the square is the Earth.
- This circle cuts the vertical axis at
*M*, with*MQ*a remainder. *Q*=*Rm*= Radius of the Moon.- With center
*Z*, a second, concentric circle, with radius*ZQ*(the radius of the Earth AND the radius of the Moon added together) is circumscribed around the circle that represents the Earth. - This new circle (radius =
*ZQ*) and the square around the Earth,*GKBN*, are very nearly equal in perimeters (close enough to wonder about Nature's intentions!).
Of note is that our earth/moon relationship is the only one in our solar system that contains this unique golden section ratio that "squares the circle". Along with this is the phenomenon that the moon and the sun appear to be the same size, most clearly noticed during an eclipse. This too is true only from earth's vantage point…No other planet/moon relationship in our solar system can make this claim. If the base of the Great Pyramid is equated with the diameter of the earth, then the radius of the moon can be generated by subtracting the radius of the earth from the height of the pyramid (see Figure 16).
[1] The Great Pyramid
is so massive that a plumbline will not hang straight down when
near the pyramid but will swing toward the structure. Cf. Tompkins,
[2] One study expressing
such scepticism is George Markovsky, "Misconceptions about
the Golden Ratio", [3] As most of
us are aware, there are volumes of writings and studies that
address the values of [4] For a thorough
text on the measures of the Great Pyramid, cf. the Appendix written
by Livio Catullo Stecchini in Tompkins, [5] Cf. Schwaller,
[6] Cf. Euclid, [7] Cf. Euclid,
Book XIII, proposition 8. [8] Robert Lawlor addresses the symbolism of these generative
powers of [9] For excellent
reading and for references into the pyramids of the Teotihuacan
Complex, cf. Tompkins, [10] I've been
working with the root- [11] An "eye"
is where two or more lines intersect. These points are valuable
in the generation of more complex geometric relationships from
simple initial constructions. The more they are used, the more
they create new eyes, and they are quite similar to fractals.
In traditional geometry, where these diagonals cross in the middle
is known as the arithmetic center, rather than a geometric one,
because it is one half the distance from all four corners, and
it demonstrates simple equality. This particular eye [12]Additionally,
in the following drawings, it should be kept in mind that we
are performing the function of multiplicity and regeneration
by placing two root- [13]The deviation
is, curiously, 3. 058…high (like our [14] [15] Both Plato's
Budge, Wallis E. A. Cook, Theodore Andrea. Euclid. Erlande-Brandenburg, Alain. Frascari, Marco, and Livio Volpi Ghirardini,
"Contra Divinam Proportionem", Harleson, Hugh. "A Mathematical Analysis of Teotihuacan", Mexico City: XLI International Congress of Americanists, October 3, 1974. Garland, Trudi Hammel. Hall, Manly P. Hambidge, Jay. Hancock, Graham. Ivimy, John. Lawlor, Robert. Mann, A. T. Markowsky, George. "Misconceptions about
the Golden Ratio." McClain, Ernest G. Michell, John. Michell, John. Michell, John. Plato. Schneider, Michael. Schwaller de Lubicz, Rene A. Schwaller de Lubicz, Rene A. Schwaller de Lubicz, Rene A., Thompson, D'Arcy Wentworth. Tompkins, Peter. Tompkins, Peter.
Nova Online:Pyramids The Great Pyramid at Khufu Encyclopedia Smithsonian's Egyptian Pyramid Page Egyptian, Pre-Colombian and Modern Pyramids Mexican Pyramids Keys to the Pyramids Teotihuacàn Pyramid of the Sun
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