A Comparative Geometric Analysis of the Heights
and Bases of the Great Pyramid of Khufu and the Pyramid of the
Sun at Teotihuacan
667 Miller Avenue
Mill Valley, CA 94941 USA
back into the murky mysteries of ancient times, there are reminders
of past glories in the art, architecture, and design of our ancestors,
and, in the number systems they employed in those designs. These
number systems were clearly expressed in the geometry they used.
Among these works are the mammoth pyramids that dot the Earth's
surface. Accurate in their placement as geodetic markers and
mechanically sophisticated as astronomical observatories, these
wonders of ancient science stand as reminders that our brethren
of antiquity may well have known more than we think. In our attempts
to understand and decode these objects of awe, we realize that
the winds of time and the ignorant hands of humanity have eroded
the precise measurements and canons that were infused into these
monuments. By their sheer magnitude ,
the pyramids tell us that their builders
clearly wanted future civilizations to not only notice them,
but to also investigate them in an attempt to find out what knowledge
these masons and priests possessed regarding the world and the
universe; and although the precision of the structures may be
missing, we can still see the intentions through the geometry
that remains. This study was undertaken with that in mind.
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
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.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.
DIMENSIONS OF THE GREAT PYRAMID AND THE PYRAMID OF THE
My initial investigation was focused
on a verification of the base/height measures of the Great Pyramid
(the Pyramid of Khufu, or Cheops in Greek) at Gizeh, Egypt, and
the Pyramid of the Sun at Teotihuacan, Mexico, and the precision
of each with regard to pi to determine if one was more
accurate than the other. I was quite content to use rational
approximations for pi ,
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.
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.
Pyramid of the Sun
Angle of slope
The rectangles generated from the height of each pyramid and
its ½ base, when the ½ base is considered to be
Pyramid of the Sun
1.2738853... : 1 (deviates by 0.15 % from the 1.2720196
value for the square root of phi)
.6369339... : 1 (deviates by 3.06 % from the 0.618034
value for the reciprocal of phi)
The ratio of the base perimeter to the height:
Pyramid of the Sun
6.2800001... : 1(deviates by 0.05 % from the 6.2831853
value for 2 x pi)
12.560171... : 1 (deviates by 0.05 % from the 12.566371
value for 4 x pi)
The ratios of the bases to one another:
Great Pyramid/Pyramid of the Sun
1.0308101... : 1
Pyramid of the Sun/Great Pyramid
0.9701107... : 1
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.
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.
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.
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
- 4 divided by root-phi is approximately equal to pi
(4/1.2720196 =3.1446055, a percentage deviation of less than
- 5/6 of phi squared is approximately equal to pi
(6/5 x 2.618034= 3.1416408, a percentage deviation of less than
- 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 pi, phi, and root-phi
in an elegant and mysterious way, and it is far too subtle to
be an accident. It is a construction that is used to raise the
elevation of the Great Pyramid that is based on the golden section,
a construction widely known to us today. There is, however, a
curious and minute measure that is also generated, a small function
generated from the original, that yields the unique relationship
with the Great Pyramid and the Pyramid of the Sun. What is most
intriguing is that the construction involves the more mysterious
square root of phi, and not its more well known relative, phi. But
let us now turn our attention to these constructions that connect
the Great Pyramid to the Pyramid of the Sun..
PHI AND ROOT-PHI
us begin with phi, the golden section ratio (Figure
1). This ratio is the division
of a segment or a magnitude into what Euclid termed "mean
and extreme ratio", that is, a division such that the lesser
is to the greater as the greater is to the total length. It is also found in the pentagram and
pentagon as the ratio between a side and a chord; if the side
is equal to 1, the ratio is equal to 1.6180339: 1.  The "Divine
Proportion", as it is sometimes called, can be found throughout
nature: in the bones of the fingers, hand, and forearm, the location
of the navel and where the fingertips touch the thigh in relation
to the height of the adult human body; in the spiral seed patterns
in a sunflower or daisy; in a spiral nebula light years from
the Earth; in the curvature of certain seashells. Geometrically,
the spiral nebula and the nautilus shell differ only in size,
because each exhibits a logarithmic spiral based on the same
golden section and the square root of the golden section numbers.
The golden section is one of the more well-known ratios that
have been used throughout history in the making of things calling
for certain types of harmony and proportion.
The golden section ratio can be constructed
in several ways. The half-diagonal of the square and the diagonal
of the double square can each be used. In Figure
2, KB can be a segment
of any length that we will call 1. By constructing a double square
AKBL on this length, the golden section of it can be found
at G. The specific steps are:
1.Draw any length KB.
2.Divide the length KB in half. ( Use a measure, a bisector,
or the Vesica Piscis. )
3.These two lines, KO and OB, are the short sides
of a double square ( root-4 ).
4.Construct the double square.
5.Draw a diagonal KL.
7.Taking the compasses, put the pin in L, and open to
B, drop LB onto the diagonal KL at q.
8.With the pin in K, open the compasses to q, and
rotate Kq to KB at G.
G is a golden section of KB, and BG : GK
:: GK : KB = 1/phi : 1 ::
1 : phi. The simplest
way is to generate the ratio is through its better-known relative,
the golden section rectangle (Figure
1.Draw square AKMZ.
2.Locate the midpoint of a side of the square AK at H.
3.The half-diagonal of the square, HM, rotated through
the arc MR to the vertical position to become HR.
Rectangle ARSZ is a golden section rectangle. Point
K is also at a golden section of AR. This rectangle
is composed of a square, AKMZ, and KRSM is also
a golden section rectangle. AZ : 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
1.Draw golden section rectangle ARSZ.
2.It is next required that the long side of the rectangle ARKZ,
that is, AR, be rotated through arc RQ to point
Q on the opposite side, SZ, of the rectangle.
The triangle that is formed, AQZ,
is sometimes referred to as the "Triangle of Price",
after W. A. Price (who named it after himself!).
This triangle is the only right, scalene triangle whose sides
are in a geometric progression, a progression based on the square
root of the golden section: 1, root-phi, phi
. Most notably, this triangle is, as we shall see, associated
with quadrature (squaring the circle) and the relationship between
the sizes of our Earth and moon. The simplicity of these constructions
conceal the power and complexity that underlies the geometry.
The angle of slope for the GP Pyramid is very nearly
degrees or 51 degrees 49' 38''. An interesting
aside is the fact that the 7-sided figure, the heptagon, has
an angle from its center that subtends an angle of 51 degrees
25' 42'', a mere 0 degree 23' 55.3'' difference. Not perfect,
but very, very close.
GEOMETRIC CONSTRUCTION OF THE PYRAMIDS
Now that we are familiar with the bases,
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
- 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
(.5) the side of the square (0.5 + 1. 11803
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-phi
rectangle, having the golden section symbolically growing as
this root as represented by the diagonal of this new rectangle.
It is the root that grows through this ratio.
The underlying truth is that geometry appears to be a hard and
mechanical device, but under the surface, there is an occult
and earthy organic quality that is manifested either by the life
force, or somehow through the geometry itself. It is in 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.)
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.
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.
1. From the previous drawing, using points
p3, draw through both p3 points to go through p4,
and draw kj, a side of the base of the Pyramid of the
Sun, parallel with KJ.
2. Continue this process around the other three sides. There
are two ways to assist in this step.
3. Draw a second circle inside the first circle.
4. By using the two diagonals of the squares, the 45-degree lines
will enable you to make the 90-degree turns to the other three
sides, and to keep the second square parallel to the first.
5. Complete the second square, bkjl, parallel to BKJL.
6.This second, inner square is the base of the Pyramid of the
Sun, in plan view.
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.
there are two constructions that will accomplish this. They are
unconnected, but both yield the same result. They are somewhat
quirky, and if you were not looking for them to generate, they
would both be easily missed, perhaps like the Platonic Year. Let us
look at Figure 8:
- 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. 618
- 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!
shows the division of the area into the reciprocal, and how it
generates point q, and that this generates three root-phi
rectangles (APRZ, PKrq, rqMR) within the
fourth and master rectangle, AKMZ.
In Figure 10 and Figure
11, the "2. 058
rectangle has been isolated. In Figure 10, the Rule of Thales
regarding points on the semi-circle and the 90-degree angle that
is formed by drawing from that point q to the ends of
the diameter KNM. It can be seen here that the "2.
" does not contain this semi-circle, because
point N lies outside the circumference; if it were a double
square, it would contain it. The Rule of Thales has been evoked
to illustrate the function of the diagonal/reciprocal relationship,
that of the crossing of the two at 90 degrees. In this instance,
the angle is at the very end of diagonal qM, with qK
as the reciprocal. In Figure 10:
- PKrq = qrMR = root-phi
- PKrq is the reciprocal rectangle to the master rectangle
- 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 
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.
following constructions and points are of the same nature. They
are not present in the master rectangle, until a detailed and
complex grid is developed. The two bases we have just seen are
the relative sizes of the two pyramids. What an ingenious way
to vary the two, with a circular curve cutting the slope.
Now we will look at the key constructions for the
Pyramid of the Sun. In Figure
12 and Figure
13, we see the elevation of the
Great Pyramid, AQL. The golden section length AR
is rotated through arc RR2 and stops on the base AL
at R2. Two root-phi rectangles are placed vertically,
sharing a vertical long side ZQ and having their diagonals
phi when their short sides are 1. The total base is now
2, with the two root-phi rectangles placed side-by-side.
The other diagonal ZJ is drawn in AQJL. Where this
arc cuts the base at R2, it divides the half-base ZL
of the Great Pyramid into a golden section relationship and
creates a golden section (geometric) progression across the entire
base. That is:
- LR2 : R2Z :: R2Z :
- 1/phi : 1 :: 1 : phi
- LZ = ZA = 2
Triangle AQL is the only isosceles triangle that can
do this. As it rotates, it cuts the second diagonal ZJ
at p1, at the precise point needed to obtain the height
of the Pyramid of the Sun. This point p1 is extremely
close to the center of the rectangle where the two diagonals
cross, point p2.
Next, we will construct
the height of the Pyramid of the Sun from point p1 (Figure 14
and Figure 15).
1. Draw the other diagonal, ZP, into
root-phi rectangle APQZ.
2. Draw the arc JG to generate a second p1 in APQZ.
3. Draw the midlines to AP and LJ at t1p2
4. Draw a parallel to t1t2 from points p1 through
Point O marks the elevation of the Pyramid of the Sun.
The difference in measurement of the drawing's construction and
the actual pyramids between the dead centers at p2 and
O to the height ZQ in the construction is 1 / 81.
This is the very lovely and amazing decimal, 0. 01234567
difference between the two heights of the two pyramids is one
part in approximately eighty-one and a tenth, 1 / 81.1. As a
decimal, it is, 0.01233045
Looking at the percentage difference
between my construction's measure, 1 / 81, to the measures given
by the quoted sources, which is 1 / 81.1 yields a difference
of, 0.0015%, an insignificant deviation.
method for finding the height of the Pyramid of the Sun (Figure 16)
requires the same double square root of the golden section compound
rectangle system used in Figure 14. In this second method, the
slightly smaller base of the Pyramid of the Sun, GN, must
be drawn from Figure 7. Points G and N on the base
that are needed.
1. Draw a line from point G to point
J, a corner of the root-phi rectangle on the right.
(Note that if the line were drawn from point A, a corner
of the root-phi rectangle on the left, the line would
cut ZQ at its mid-line. This procedure yields point O
on ZQ, lower on ZQ to the same point as was yielded
in Figure 14.)
2. Repeat this procedure from point N.
Lines GO and ON are the slope of the Pyramid
of the Sun.
In each of these two methods, the
height of the Pyramid of the Sun is dependent on the complete
Great Pyramid construction for its measure.
in the sequence of drawings, we complete the elevation for the
Pyramid of the Sun (Figure 17). (This isosceles triangle is not intended
to be the correct profile of the Pyramid of the Sun, which has
stepped courses, but is instead a representation of the pyramid's
height, as found by drawing through the edges of the stepped
courses of the slope, and width.)
- 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.
Had the apex gone to the exact half way point (designated
by the base measure GN, which is about 97% of the Great
Pyramid), the rectangles would have been golden section rectangles.
Although I am uncertain about the reasons for this curiosity,
I do want to present it for future discussion. Perhaps this "slight"
of the golden section demonstrates the builders' faithfulness
to the higher plan that involves both. The builders of these
structures may have been so exacting in their work that the truth
may lie in a deliberate "error"!
The Master Diagram (Figure
18) is the compilation of the
individual constructions to show the overall, complete relationship
between the two pyramids. Let us review its components. This
construction shows the development of the golden section ratio,
the progression to its square root, which yields the Triangle
of Price, AQZ. The bilateral symmetry of a second Triangle
of Price ZQL yields the elevation of the Great Pyramid.
As the long side AR of the golden section rectangle is
rotated over to its opposite side to generate the square root
of phi at Q, AR continues down to the horizontal
position at R2 as AR2. As the arc is swung, it
cuts the diagonal ZJ of the square root rectangle ZQJL
at P1. P1 yields the height of the Pyramid of the
Sun at O. This height can also be generated by a diagonal
line from the base vertex, G, to a vertex of the square root
rectangle at J, and cutting the vertical axis QZ at O.
The slight difference, MP4, is equal to the differences
in the base perimeters. Because the chances of this coincidence
occurring are quite remote, it appears that perhaps someone had
done the construction originally and intentionally used this
slight variation in the two pyramids. By studying the complete
drawing, the beauty and elegance of the Master Construction comes
through, and the connection between the two becomes much clearer.
THE SEARCH FOR MEANING: THE EARTH, THE MOON AND THE SQUARE
ROOT OF THE GOLDEN SECTION
The square root of the golden section,
1.2720195..., is a quite remarkable number and construction.
I learned of it years ago when studying spirals and the way they
could be generated by various ratios, angles and lengths. After
repeated failures and several shells purchased in order to substantiate
the claims that the chambered nautilus was generated from the
golden section rectangle, I finally realized that the correct
rectangle for framing the shell is root-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
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!
- 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
- 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
- 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).
of all the number systems and numerical relationships that could
have been used, the Egyptians selected the golden section ratio
and its square root in the building of the Great Pyramid. The
Pyramid of the Sun also contains a sophisticated and subtle "infusion
by association" of the same ratio. Through a specific series
of points within the construction of the Great Pyramid's elevation,
the elevation of the Pyramid of the Sun can also be generated:
the two elevations are linked through this precise geometry.
That two different architects would use points in the geometric
constructions in the singularly unique way I have discussed seems,
to me, most improbable, perhaps impossible, for they serve neither
a functional purpose, being totally unnecessary for construction
purposes, nor a specific spiritual, symbolic, or philosophical
purpose. However, it appears intentional. And if it was intentional,
was it also intentionally mysterious? After examining the geometric
analyses, it becomes evident that the two structures fit together
as one geometric construction. But unless one is familiar with
the various golden section and square-root-of-the-golden-section
constructions, this particular and unique relationship could
be easily missed.
The insights afforded by geometric
analysis are valuable in gaining a deeper awareness of the makers'
intentions. In this particular instance, the insights led to
even bigger questions. Were these two pyramids somehow related
intimately in other ways than by the geometry inherent in both?
Why would two pyramids be so very nearly in a 1-to-2 relationship
in their heights, yet not "exactly", or "not at
all"? After examining the constructions presented here,
it seems too great a coincidence that they are "almost exactly"
alike but not related.
Although the two pyramids
were separated by the Atlantic Ocean, in vastly different locations
and cultures, is it possible that the two pyramids were built
by the same architect or architects, either directly, or by courier
with instructions? Was the ocean transversed, or was there a
continent that connected the Americas with Europe and Africa,
and inhabited by intelligent and gifted people?[
However, even though the two constructions
fit together in such a unique and curious way, this is not undeniable
scientific or mathematic proof that the two pyramids were designed
and executed by the same architects. Much more would need to
be done. Documents on the subject would be grand!
I write, it is my understanding from a number of sources that
parts of the Great Pyramid were closed for a few years for the
purposes of cleaning the graffiti from the walls and installing
a new air conditioning and ventilation system. In addition, rumors
persist that ancient texts exist at the Gizeh complex. Modern
underground sonar studies indicate that there is a large room
or space under the paws and belly of the Great Sphinx, and that
there is a second "gallery" that extends from the Sphinx
to a large hall or room directly under the Pyramid along an east/west
axis. The rumor advanced with these stories is that, under the
cover of the edifice itself, away from public view, workers have
begun the descent inside, straight down from the apex of the
pyramid; at the specific depth indicated by a scan of the area
already undertaken, they will turn due east toward the Sphinx
. Their intent is to reach this corridor and to follow it to
the area recorded on the sonar. The echoes indicate that there
is a hollow area under the Sphinx, and officials within the government
have remained tightlipped and have refused further study and
investigation to be conducted by anyone outside their auspices.
The commonly held belief is that this hall is the fabled Hall
of Records, and contains writings and documents that explain
the origins and history of mankind on the planet. Perhaps there
are documents that link these two pyramids!
Although there can be no final declaration
for or against the hypothesis I have presented about these two
pyramids being linked by a common builder or building culture,
I hope that I have at least opened the possibility for further
work on the subject. Although ragged and worn, each building
stands as a memorial to something that was marvelous, and is
still mysterious. They serve to remind us also that the human
body, mind and spirit have no boundaries.
 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,
Secrets of the Great Pyramids, pp. 84-85, where Tompkins,
discussing the measurements taken by Piazzi Smyth, writes "To
obtain the correct latitude of the Great Pyramid without having
his plumb line diverted from the perpendicular (italics are
mine) by the attraction of the huge bulk of the Pyramid, Smyth
made his observations from the very summit; there the Pyramid's
pull of gravity would be directly downward". return to text
 One study expressing
such scepticism is George Markovsky, "Misconceptions about
the Golden Ratio", The College Mathematics Journal,
23, 1, (January 1992), pp. 2-19. return
 As most of
us are aware, there are volumes of writings and studies that
address the values of pi, but for those who are not acquainted
with the literature, I recommend Petr Beckmann's book, A History
of Pi, and any one of the books written by John Michell on
the subject of ancient measures, especially, The New View
Over Atlantis and The Dimensions of Paradise . Michael
Schneider was kind enough recently to draw my attention to another
fraction used in antiquity: 355/113. Also, as John Ivimy points
out in his The Sphinx and the Megaliths (Chap. 9, p. 116),
A. Thom's studies of ancient and Megalithic sites indicate that
the geometers of the era abhorred incommensurables. I am in total
agreement with this view. As I have already mentioned, the focus
needs necessarily to be on the intentions of the builders, and
even on the cleverness and intuitive powers of the ancient mind
to solve an irrational problem with whole numbers and fractions
contrived from natural integers. Most importantly, it has been
shown by a number of writers and scholars ( including R. A. Schwaller
de Lubicz, Robert Lawlor, and Peter Tompkins ) that what is more
significant is not in the accuracy of pi, but rather,
its connection with the square root of the golden section, root-phi,
in the formula: 4 / pi = root-phi. This equation
leads to the problem of Quadrature, the Squaring of the Circle.
return to text
 For a thorough
text on the measures of the Great Pyramid, cf. the Appendix written
by Livio Catullo Stecchini in Tompkins, Secrets of the Great
Pyramid. I also recommend Tompkins' history of the great
tradition of measures that have been done on the Great Pyramid
and the Gizeh Complex, and the measurers/ geometers who did them.
return to text
 Cf. Schwaller,
The Temple of Man, Chapter 5, "Foundations of Pharaonic
Mathematics", pp.88-125. return
 Cf. Euclid, The Elements, Book VI, definition
3. return to
 Cf. Euclid,
Book XIII, proposition 8. return to
 Robert Lawlor addresses the symbolism of these generative
powers of phi in Sacred Geometry, pp. 44-64. return to text
 For excellent
reading and for references into the pyramids of the Teotihuacan
Complex, cf. Tompkins, Mysteries of the Mexican Pyramids,
Chapter 18, "Mathematical Extrapolations", pp.241-263.
Peter Tompkins does an excellent job of weeding and clearing
up Hugh Harleson's extrapolations. I don't agree with his measuring
schema that he calls the "hunab ", a special measure
remotely similar to Thom's megalithic yard. return
 I've been
working with the root-phi ratio for a long time and I
know these peculiar items well. By example, I will define one
of the quirky things about the root-phi function. Multiplying
root-phi and phi , we get
= 2. 058170
, an amazing and
curious number. By subtracting Unity from it, you will have the
number, 1. 058170
The frequency ratio for the 12 equal
parts, or notes, of even- tempered tuning for musical instruments
is: 1. 059463
Simply stated, a string will vibrate when
plucked, and will sound some note. Then, if the string is shortened
or lengthened to the next note up or down, it will carry the
original note with it to the next note, by the note and dividing
(lower) or multiplying (higher) by the constant, 1. 059463
Trudi Hammel Garland, Math and Music, pp. 40-43). The
deviation, minus the One, is - 0. 114229
% low. What is
the significance of subtracting Unity? The cutting off of squares,
as Hambidge points out in Dynamic Symmetry, was a design
practice of the Greeks. It is also used by the Greek Mathematicians,
with whom the artists and designers collaborated, to study "compound
ratios" that were generated from the addition and subtraction
of the square, One. Following then, if a rectangle be drawn that
has a short side equal to one, and the long side equal to root-phi
x phi, or 05817
and a square is cut off on the long
side, the remaining rectangle will be only about . 001
this frequency ratio. Now the rectangle will not necessarily
make music, but the geometer/artist or builder believes that
by infusing these types of number concepts into geometric shapes
that the geometry will have a certain quality, a function, to
a higher order. The second part of this example is a more subtle
and technical one in that when the "2. 058
is drawn out, or is generated by the harmonic deconstruction
of the root-phi rectangle (See Illustration Iabove), this
rectangle is created when the reciprocal is drawn. The remainder
is this rectangle. The first instinctual response is to think
that it is a root-4 rectangle, the double square, 2 to 1. There
is only a .058
difference between the two. return to text
 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 p1
is not one that would ordinarily be considered or used in a design.
It would be far more logical to simply use the arithmetic mean,
half way, of the height, if the builders are making references
to twice or four times pi in the relationship of base perimeter
to height. Rather, they used this second and far more hidden
point, p1, to build with. The masters who built the pyramids
were precise, and I find it is unlikely that this "slightly
less than half the height" was accidental. return
in the following drawings, it should be kept in mind that we
are performing the function of multiplicity and regeneration
by placing two root-phi rectangles side-by-side on the
sides. Their bases, as the Great Pyramid itself,
have the value of two, not one. return
is, curiously, 3. 058
high (like our phi times root-phi
but this time with Unity added, not subtracted!
(See note 10.) return to text
 The Curves
of Life by Theodore A. Cook is far and away the greatest
work on the subject. return to text
 Both Plato's
Timaeus and Critias speak with certitude that the
continent of Atlantis did exist. John Michell, the great British
scholar and author of dozens of books on ancient and universal
canons of measure, in his book, "The Dimensions of Paradise",
even goes into the dimensions of the fabled land, taking Plato
at his word. return to text
Beckmann, Petr. A History of pi. (New
York: The Golem Press, 1971). To order this book from Amazon.com, click
Budge, Wallis E. A. The Egyptian Heaven
and Hell (Chicago: Open Court Publishing Co., 1925).
Cook, Theodore Andrea. The Curves of Life
(New York: Dover Publications, Inc., 1979). To order this book from Amazon.com, click
Euclid. The Thirteen Books on the Elements.
Thomas L. Heath, trans. (New York: Dover Publications, Inc.,
1956). To order this
book from Amazon.com, click
Erlande-Brandenburg, Alain. Cathedrals
and Castles: Building in the Middle Ages (New York: Harry
N. Abrams, 1995). To
order this book from Amazon.com, click
Frascari, Marco, and Livio Volpi Ghirardini,
"Contra Divinam Proportionem", Nexus II: Architecture
and Mathematics, Kim Williams, ed. (Fucecchio,Florence: Edizioni
dell' Erba, 1998).
Harleson, Hugh. "A Mathematical Analysis
of Teotihuacan", Mexico City: XLI International Congress
of Americanists, October 3, 1974.
Garland, Trudi Hammel. Math and Music
(Palo Alto: Dale Seymour Publications, 1995). To order this book from Amazon.com,
Hall, Manly P. The Secret Teachings of
All Ages (Los Angeles: The Philosophical Research Society,
1988). To order this
book from Amazon.com, click
Hambidge, Jay. The Elements of Dynamic
Symmetry (New York: Dover Publications, 1967). To order this book from Amazon.com,
Hancock, Graham. Fingerprints of the Gods
(New York: Crown Trade Paperbacks, 1995). To order this book from Amazon.com, click
Ivimy, John. The Sphinx and the Megaliths.
(London: ABACUS, 1976).
Lawlor, Robert. Sacred Geometry. (London:
Thames and Hudson, 1982).To
order this book from Amazon.com, click
Mann, A. T. Sacred Architecture (Rockport:
Element Books, 1993).
Markowsky, George. "Misconceptions about
the Golden Ratio." The College Mathematics Journal,
vol. 23, no. 1, (January 1992), pp. 2-19.
McClain, Ernest G. The Myth of Invariance
(York Beach: Nicolas-Hays, 1976). To order this book from Amazon.com, click
Michell, John. The Dimensions of Paradise
(London: Thames & Hudson, 1988).
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Mystery of Glastonbury, (London: Gothic ImagePublications,
Michell, John. New View Over Atlantis
(London: Thames & Hudson, 1983). To order this book from Amazon.com, click
Plato. Timaeus and Critias. Desmond
Lee, trans. (London: Penguin Books, 1965). To order this book from Amazon.com, click
Schneider, Michael. The Beginners Guide
to Constructing the Universe (New York: Harper Collins, 1995).
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from Amazon.com, click
Schwaller de Lubicz, Rene A. The Egyptian
Miracle (Rochester: Inner Traditions International, 1988).
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from Amazon.com, click
Schwaller de Lubicz, Rene A. Sacred Science
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Tompkins, Peter. Mysteries of the Mexican
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SITES ON THE WWW
The Great Pyramid:
Schematics and Photos
Great Pyramid at Khufu
Smithsonian's Egyptian Pyramid Page
Pre-Colombian and Modern Pyramids
to the Pyramids
of the Sun
ABOUT THE AUTHOR
Mark Reynolds is a visual artist who works primarily in drawing,
printmaking and mixed media. He received his Bachelor's and Master's
Degrees in Art and Art Education at Towson University in Maryland.
He was awarded the Andelot Fellowship to do post-graduate work
in drawing and printmaking at the University of Delaware. For
the past decade, Mr. Reynolds has been at work on an extensive
body of drawings, paintings and prints that incorporate and explore
the ancient science of sacred, or contemplative, geometry. He
is widely exhibited, showing his work in group competitions and
one person shows, especially in California. Mark's work is in
corporate, public, and private collections. Mark is also a member
of the California
Society of Printmakers
(six of his images can be found on their website by clicking
on "Galleries" then scrolling down to Mark Reynolds
under "Artist Member Porfolios), the Los Angeles Printmaking
Society, and the Marin Arts Council.
A born teacher,
Mr. Reynolds teaches sacred geometry, linear perspective, drawing,
and printmaking to both graduate and undergraduate students in
various departments at the Academy of Art College in San Francisco,
California. He was voted Outstanding Educator of the Year by
the students in 1992.
Additionally, Reynolds is
a geometer, and his specialties in this field include doing geometric
analyses of architecture, paintings, and design. He presented,
New Geometric Analysis of the Pazzi Chapel", at The Nexus 2000 Conference in Ferrara, Italy.
Mr. Reynolds is also contributing editor for the "Geometer's
in the Nexus Network Journal. He lives with his wife and
family in Mill Valley, California.
The correct citation for
this article is:
Reynolds, "A Comparative Geometrical Analysis of the Heights
and Bases of the Great Pyramid of Khufu and the Pyramid of the
Sun at Tetihuacan", Nexus Network Journal, vol. 1
( 1999), pp. 87-92. http://www.nexusjournal.com/Reynolds.html
Copyright ©1999 Kim Williams
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