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A Critique of Graham Hancock’s
Forced Numerical Relationship
between the Great Pyramid of Giza
and Earth’s Dimensions
Thomas W. Schroeder, 7 November 2019
Graham Hancock regularly draws attention to what
he considers mystical relationships between the
Great Pyramid of Giza and the radius, circumference,
and axial precession of the earth (Figure 1, shown on
page 6, illustrates axial precession). Proponents of
these “mystical” relationships contend, in addition to
existing in the first place, that the relationships must
be purposeful and therefore provide direct evidence
of advanced capabilities in technology, mathematics,
and precise astronomical observing techniques that
scholars have long asserted were not available to
humans when the pyramids were constructed. Others
may find these esoteric connections between the
Great Pyramid and the earth to be dubious given
many natural observations that explain the corpus of
geometric semblances without ascribing a precise
knowledge of π to its builders, or contending that the
Great Pyramid was coded with some of earth’s key
parameters that Hancock refers to as earth’s “cardinal
dimensions.”
The explanations offered here extend inherently from
simple and reasonable pyramidic design techniques,
though the explanations themselves are sometimes
technical. While each concept is basic, it can be
convoluted to explain them in writing. It is quite easy
for Hancock to concoct fascinating claims and appeal
to unknown advanced intellects for justification, but
cumbersome to demonstrate how his superficially
profound relationships evaporate when relying on a
range of actual measurements and observations. This
is especially true when so many people prefer being
amazed by mystical accounts over genuine
assessments of human ingenuity.
In this 5-minute video, Hancock summarizes some of
his fantastic claims (Hancock 2017).
Though Hancock is widely popular, he is not alone in
publishing revelations of hidden codes and lost
ancient secrets. He is part of a troupe of Western
esotericists that has grown in popularity over the past
30 or so years. This contemporary esotericism,
though likely rooted in Pyramidology and the
Christian Identity Movement, appears generally more
focused on a pseudoarchaeological message of
ancient and lost scientific knowledge, while the
meme that divine pyramidic codes link English
heritage to Biblical racial identities, has lost favor for
various reasons (Keach 2011).
Prominent mentors of this esoteric movement
include Eckhart R. Schmitz, Edward J. Nightingale,
Robert Bauval, Adrian Gilbert, Zecharia Sitchin and
others. Schmitz and Nightingale particularly circulate
various hidden numerical relationships that they
both perceive or construct. This is seen in Schmitz’s
book; “The Great Pyramid of Giza: Decoding the
Measure of a Monument”, details of which are
included on Hancock’s website (Schmitz 2012), and in
Nightingale’s book, “The Giza Temple”, a summary of
which Hancock again promotes on his website
(Nightingale 2015).
This critique focuses on Hancock’s and others’ claims
that the Great Pyramid’s proportions contain hidden
codes, and that those codes demonstrate proof that
the ancient builders were aware of earth’s precession
and circumference and capable of calculating both to
a degree of accuracy inconsistent with ancient
knowledge and technology (Hancock 2016).
Basic geometry and mathematics are used to discuss
π, Ø, precession, circumference, radii of the earth,
etc., and to rebut claims of alien or ancient hyper-
advanced human contributions to the design
characteristics of the Great Pyramid. However, the
practical logistics of how blocks were quarried,
carved, moved, polished, and stacked is ignored since
those aspects are regularly addressed and

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demonstrated by others (“Egyptian Pyramid
Construction Techniques.” 2019). Instead, attention is
focused on specific observations that indicate the
intrinsic, trivial, or nonexistent relationships shared
between the Great Pyramid of Giza and the earth.
Hancock Being Hancock
For those new to Hancock, here is an example of his
penchant for hyperbole, esotericism, and general
gravitation towards pseudoarchaeology. In this
excerpt from the video link above, Hancock speaks to
the alignment of the Great Pyramid:
“The Great Pyramid is locked in to the cardinal
dimensions of our planet. The Great Pyramid is
targeted on true north, within three sixtieth of a single
degree. Now, no modern builder would create a large
building and add onto his or her shoulders the
additional burden of aligning it to true north within a
fraction of a single degree. They just wouldn’t get it;
they wouldn’t understand why it was important to do
that” (Hancock 2017).
One doesn’t need to understand algebra, the earth’s
geometry, or methods of astronomical observation to
recognize Hancock’s desire to sensationalize and
mysticize pyramidic construction techniques upon
hearing that statement.
Firstly, monument builders, both modern and
ancient, are fully capable of understanding why they
might align public or private architecture to any
specified azimuth or any specific placement, both
from a purely functional perspective; i.e. building
locations and orientations are typically specified, as
well as a symbolic perspective; i.e. true north, due
east, or “towards Mecca” all have natural interpretive
understandings that might appeal to an architect or
group of people. Symbolic affinities are especially
appealing for monuments but can be manifested in
any architectural undertaking.
Secondly, constructing a building’s location and
alignment with great precision would be trivial given
surveying technology and methods that exist today.
Modern equipment measures the propagation of
electromagnetic waves of varied wavelengths and can
be as precise as 0.5 to 5 mm at distances of up to 3 km
(“Electronic Distance Measurement Instrument”
2017).
Despite deep admiration for the construction
techniques and precision of the ancient Egyptians,
suggesting that builders today would struggle to
reproduce such an alignment is either foolish or
disingenuous. Drawing attention to the impressive
accuracy achieved by the pyramid builders is
certainly a legitimate way to recognize them for their
ingenuity, religious like attention, and remarkable
application of technologies available to bronze age
monument builders. However, this recognition can
be offered without suggesting that surveyors and
builders today couldn’t recreate or surpass ancient
accuracy and precision.
Geometry, Precision, and Purpose
From the video: “… to incorporate into its dimensions
the dimensions of our planet. I don’t want to get too
numerical or possibly even boring here, but if you take
the height of the Great Pyramid and multiply it by
43,200, you get the polar radius of the earth. And if you
measure the base perimeter of the Great Pyramid
accurately, and multiply that measurement by 43,200,
you get the equatorial circumference of the earth ...
… and the scale is not random. The number 43,200 is
derived from a key motion of the earth, which is called
the precession of the earth’s axis. The earth wobbles on
its axis very slowly at the rate of one degree every 72
years. And 43,200 is a multiple of 72. In fact, I think it
is 600 times 72” (Hancock 2017).
In the above paragraph, Hancock emphasizes that the
primary dimensions of the Great Pyramid directly
equate to the earth. He marvels at these relationships
as if to suggest that these relationships are so mystical
they reveal knowledge of the builders that cannot be
explained by the known technology of ancient Egypt
alone. In other interviews and writings Hancock
explicitly claims these relationships demonstrate the
existence of unknown hyper-advanced Pleistocene
civilizations (Hancock 2016).
Note that criticizing Hancock’s specific claims that
the ancient Egyptians coded precession and earth’s
dimensions into the Great Pyramid does not imply a
strict denial that the ancient Egyptians understood
precession and made some estimate of it, or that they
built the Great Pyramid to some purposeful size they
felt was harmonious to either the earth or nature.
This critique is simply intended to show that
Hancock’s support for these claims is entirely forced
or imagined.

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In the sections that follow, these four main ideas are
demonstrated to rebut Hancock’s appeal to hyper-
advanced civilizations:
1) Observed relationships between the
dimensions of the Great Pyramid and the earth
can be completely natural: Any pyramid
constructed with the geometry of a simply circle in
mind is naturally comparable and proportional to any
circle or sphere. This is easy to show.
2) 43,200 does not indicate precession: The scalar
43,200 is not suggestive of a knowledge of precession
and has no direct association with precession or any
of earth’s other parameters.
3) 43,200 is probably not the Great Pyramid scalar
if such a scalar relating the Great Pyramid to the
earth even exists: Hancock (and others such as
Eckhart) cherry pick their measurements from a wide
range of possibilities, to demonstrate that the ratio of
the Great Pyramid to the earth is precisely 1:43,200.
Even if the ancient Egyptians intended for the Great
Pyramid to be scaled to any dimension or natural
movement of the earth, there is no evidence that the
scale is precisely 1:43,200.
4) Alternative (including more precise) values for
precession and the size of the earth can’t be
demonstrated: After refuting Hancock et al. claims
that the ancient Egyptians incorporated the value of
25,920 years precession (25,920 is a fairly precise
value of precession and, if understood, would indicate
impressive technology for ancient astronomers), the
earth’s equatorial circumference, and the earth’s
polar radius into the Great Pyramid’s dimensions, it
is demonstrated that it is also impossible to prove
that even more accurate and precise values were
coded into the Great Pyramid. This added effort is
offered since a reader might consider that Hancock et
al. are correct to recognize that the ancient Egyptians
coded both precession and earth’s dimensions into
the Great Pyramid, but their error is that they failed
to realize that the Great Pyramid coded greater
precision than Hancock recognizes. The extended
analysis is presented in Appendix E.
Evaluating the Relationship between the
Great Pyramid, Earth’s Polar Radius, and
Earth’s Equatorial Circumference
To begin with, Hancock claims that the height and
base perimeter of the Great Pyramid equate to both
the radius and circumference, respectively, of a circle
when multiplied by the same constant, 43,200. The
circle, in this case, is a cross section of the nearly
spherical earth. The constant 43,200 will be
investigated and discussed in later sections, but for
now discussion will focus on why the height and
perimeter of the Great Pyramid can be related to the
earth when each are multiplied by some scalar
constant.
If a pyramid is purposefully constructed to
incorporate the geometry of a circle; that is, if its
height is selected to be equivalent to the radius of a
circle with a circumference equal to the pyramid’s
base perimeter, or vice versa, if the base perimeter is
selected based on the circumference of a circle with a
radius equal to the pyramid’s height, then not only
would π be naturally and systematically incorporated
into that pyramid, but those core dimensions of the
pyramid would naturally be proportional to any
circle, including those of the earth. This is true
whether the builder is aware of π or not or possesses
a precise estimate of π or not.
AXIOM 1: Any pyramid built in congruence
with the basic geometry of a circle will
incidentally contain π and be proportional to
all circles and spheres.
There are other interesting and novel explanations
for “accidentally” incorporating π into the Great
Pyramid’s construction including a method where the
base is laid out using a wheel (Yochim 2017).
Attention is drawn to the association with π because
Hancock and his ilk like to claim that π was also
coded within the pyramid in another mystical
assertion that the real and transcendental number, π,
was understood to many decimal places (Schmitz
2012). It is further shown in Appendix F that the
transcendental number Ø (commonly referred to as
the “golden ratio”) would also be intrinsically fixed
into the geometry of the pyramid designed with a
circle in mind.
If the Great Pyramid was built in such a manner it
would be comparable to a circle of any size by simply
multiplying the pyramid dimensions by a scalar. The
circle Hancock selected for comparison is the one
formed by the cross-section of the earth but about
43,200 times bigger than the Great Pyramid (Again,
more about 43,200 in later sections). Hancock
emphasizing his curiosity that the Great Pyramid

4. 4
somehow describes both the radius AND the
circumference of the earth is trivial for any pyramid
constructed with a circle in mind since the pyramid
can be related to all circles. Of course, most cross
sections of the earth are slightly different in size (and
not perfect circles), but, as will be shown, the Great
Pyramid’s dimensions still fail to align with any of
earth’s dimensions when using a single scalar and
reasonable measurements of the pyramid.
To illustrate how easy it is to associate the Great
Pyramid with a circle, some simple algebra is used
along with a very selective choice in the radius of the
earth (a range of values are available) to show, in
Appendix B, that the volume of the Great Pyramid
can also be equated to the volume of the earth using
the scalar 43,200 (though 43,200 is not a unique
solution, the process can be accomplished for many
scalars within a range of possibilities). If Hancock
himself emphasized that 43,200 also described the
volume of the earth, without explaining how
naturally volume of a sphere can be linked to any
structure that unintentionally incorporated π, or that
effort is required when choosing each input
parameter, it would appear to be yet another mystical
relationship.
Explanations have been simplified up to this point by
discussing one circle and one radius. But the earth
has more than one radius, and Hancock specifies that
the Great Pyramid’s height is specifically comparable
to the polar radius and the perimeter is specifically
comparable to the equatorial circumference. If that
distinction made by Hancock was strictly correct, and
the polar radius does equate solely and specifically to
the pyramid’s height, and the equatorial
circumference equates solely and specifically to the
pyramid’s base perimeter, that would be of great
interest. But those identities are not specifically
satisfied by any of the various earth’s radii as
explained further in Appendix A, and in the next
section. To the contrary, it can be demonstrated that
the height and base perimeter of the Great Pyramid
more likely describe the same circle which would
naturally be the case if the pyramid was designed with
the geometry of any simple circle in mind.
Evaluating Hancock’s Calculations and
the Number 43,200
Given the circumference of a circle, C = 2πr, a
pyramid can be designed with that geometry in mind,
but without understanding the precise value of π. As
was explained in the section above, this can be done
by setting the height, “h”, of the pyramid to be “r”, and
the base perimeter to be “C”. Each side of the base of
the pyramid is then C/4 or (2πh)/4.
• Equation 1: Circumference of a Circle = C = 2πr
• Eq. 2: Pyramid height = h
Setting a pyramid’s height equal to any circle’s radius
gives:
• Eq. 3: Pyramid height = h = r
• Eq. 4: Pyramid perimeter = 4 * Base side = 4*b
Setting a pyramid’s perimeter equal to any circle’s
circumference gives:
• Eq. 5: Pyramid perimeter = 4*b = (2πh)
• Eq. 6: Pyramid side = Eq. 5 divided by 4 = (2πh)/4
= (πh)/2
One can then multiply “h” of any pyramid by “k” (a
scalar) and directly relate the pyramid to a circle that
is “k” times bigger.
According to Hancock, to equate the Great Pyramid
to the earth, k = 43,200. Hancock also claims, more
specifically, that:
• Eq. 7: Great Pyramid height * k = the radius of the
earth at the poles
• Eq. 8: Great Pyramid perimeter * k = the
circumference of the earth at the equator
The earth is not a perfect sphere and therefore has
multiple implied radii in addition to multiple actual
radii. More is explained in Appendix A; but suffice to
say that if one desires to satisfy Eq. 7 and Eq. 8 with
a single scalar close to the number 43,000, one can
proceed with some confidence because of the many
radii choices available (in addition to choices in the
underlying pyramid measurements themselves given
the deteriorated state of the pyramid). Essentially,
any radius from 6,357 km to 6,400 km can be selected
as the polar radius. Likewise, the equatorial radius
can range from 6,335 km to 6,378 km, resulting in an
equatorial circumference range of 39,804 km to
40,074 km.
Continuing with Eq. 7 (with “k” = 43,200):
• Eq. 7: Great Pyramid height * 43,200 = Polar
Radius

5. 5
146.58 m * 43,200 = 6,332.26 km
Here the preferred value for the pyramid height, as it
is assumed to have existed in antiquity, is chosen. It
is now greatly deteriorated and smaller than its
original size. Much more is discussed about the Great
Pyramid size in Appendix E. However, using the
assumed height results in a polar radius that is
outside the range of possible polar radii. Conversely,
if either of the two extreme values for the polar radius
(N = 6,357 or R = 6,400) are chosen, pyramid heights
of about 148.15 m and 147.18 m, respectively, are
implied. These implied heights unfortunately fail to
form a range around the assumed original height of
146.58 m.
The radius equating to the height of the pyramid and
the constant 43,200 is 6,332.26 km and does not
match any of the many radii given in Appendix A. The
calculated value of 6,332.26 most closely matches the
meridional radius of curvature, M, at about 0 degrees
latitude (the equator). (FYI: The latitude of the Great
Pyramid is 29.9792 degrees North).
Continuing now with Eq. 8:
• Eq. 8: Great Pyramid perimeter * 43,200 =
Equatorial Circumference
230.34 m * 4 * 43,200 = 39,802.75 km (actual
circumference is 40,075 km). And 39,802.75 km
equates to an implied radius of 6,334.80 km.
In Eq. 8, the preferred value for the base side (230.34
m) of the pyramid in antiquity was used. Remember
that this method also incorporates π since that
estimate is based on the idea that the height is related
to the base as a function of π. This time the
measurement closely matches the meridional radius
of curvature at the equator and corresponds to what
Hancock stated. The difficulty with this approach is
that there is considerable variance in the possible
values of the Great Pyramid’s size that Hancock
simply resolves by fiddling with the base perimeter
until a perfect match is claimed. The actual
calculation resulted in 39,803 km versus an actual
equatorial circumference of 40,075 km).
Hancock must play a back and forth game with
numbers until he believes he has found starting
points that best fit his desired ending points. Eckhart
plays a lot with the base sides, eventually taking an
average of two measures (among dozens of choices),
and cherry picks his numerator to “show” that 43,200
solves the pyramid to earth comparison to many
decimal places.
The actual circumference of the earth, at the equator,
is 40,075 km. It can be directly seen that Hancock’s
calculation of 39,802.75 does not match. However,
since Appendix A provides ranges for the earth’s radii,
and since someone might suggest the ancient
Egyptians measured the radius at the equator, this
analysis will continue by looking at both 6,332.26 km
(the implied polar radius) and 6334.80 km (the
implied radius at the equator).
At this point it is very helpful to note that the two
methods in Eq. 7 and Eq. 8 result in two implied radii
that are close (6,332.26 km and 6334.80 km). Being
close is naturally expected if the pyramid was built
with the idea of a single circle in mind rather than
precisely coding the earth’s actual equatorial
circumference and the earth’s actual polar radius
independently. Also, π was used to calculate the
implied radius in Eq. 8, and for Eq. 7 the radius was
directly given. But what if the ancient Egyptians did
not use π? What if they used a common
approximation observed throughout antiquity, such
as 22/7? Using 22/7 in Eq. 8 instead of π produces and
implied radius 6,332.26 km, an exact match!
An exact match need not be a surprise. That must be
the case if the original height of the pyramid equates
to the radius of a single circle whose circumference is
equal to the base perimeter of the pyramid, 22/7 is
used in place of π, and the estimates of the original
height and base sides are accurate. (The assumed
original measurements are height = 280 cubits and
base side = 440 cubits. These round cubit measures
are complementary to 22/7).
AXIOM 2: For any pyramid, if the base
perimeter divided by the height is equal to 2π,
then the following two identities cannot both
be correct:
1) The pyramid height times a scalar, k,
equals the earth’s polar radius (Circle A).
2) The pyramid base perimeter times a
scalar, k, equals the earth’s equatorial
circumference (Circle B).
This is because the polar radius and the equatorial
circumference each describe two different circles, A
and B respectively. Conversely, if both the height and
the base perimeter, when multiplied by the same

6. 6
scalar, independently equate to two different
dimensions of earth; that is, if the height of a pyramid
describes Circle A, and the base perimeter describes
Circle B, then the ratio of the base to the height can
never be 2π!
Given:
• Circle A ≠ Circle B
• Circle A Radius = Pyramid Height * k
• Circle B Circumference = Pyramid Base * k
then:
• Pyramid Base / Pyramid Height ≠ 2π,
because of the following contradiction:
• Circle B Circumference / Circle A Radius ≠ 2π
It is not clear if Hancock understands these geometric
requirements, but he has stated the opposite; that the
Great Pyramid does describe π, and that it equates to
two independent dimensions of the earth (Hancock
& Bauval 1997). Even as he does, he naturally
discovers errors that he simply accepts as being
within a reasonable range, though his hypothesis
fundamentally violates the vary geometry that he
claims the ancient Egyptians were purposefully
modelling.
Appendix C summarizes calculations independent of
Hancock and based entirely on evidence. Those
calculations demonstrate how both the height and
base perimeter of the Great Pyramid likely
corresponds to a single circle. Whether that single
circle represents some aspect of the earth has not yet
been definitively shown.
The Axial Precession of the Earth and the
Search for how it Relates to a 43,200
Scalar
As the earth spins on its axis, that axis wobbles. The
wobble is somewhat uniform and creates a small
circle which is called precession. Precession exists for
all spinning disks or spheres. For the earth, one full
cycle of precession is completed roughly every 25,920
years, though 25,771.5 is the most recent modern
estimate. Hancock et al. gravitate towards 25,920 for
various reasons so this analysis will continue using
Hancock’s published value of 25,920 (Hancock 2011).
However, more precise values for precession are also
considered in Appendix E. Hancock claims that the
scalar of 43,200 also indicates direct knowledge of
precession, and that the cycle the ancient Egyptians
calculated was specifically 25,920 years. To verify this
requires verification that the Great Pyramid was built
with a purposeful scalar of 43,200, and that the scalar
also purposefully indicates the number 25,920.
Figure 1

7. 7
Hancock connects 43,200 to 25,920 primarily because
25,920 divided by 360 gives 72 and 72 is also a factor
of 43,200.
Clarification: In the video referenced above,
Hancock rounds precession to 26,000. In other
videos and publications, he specifies the more
precise number of 25,920. This discrepancy in the
video is likely conversational rounding. Hancock
does point out in the video that the number is the
product of 360 * 72. Therefore, it is assumed that
Hancock generally prefers the more precise value
of 25,920 and focus will be on analyzing that
number instead of 26,000. However, as 25,920 is
analyzed, all criticisms also apply to 26,000.
Attention is carefully called to any argument where
26,000 might be more beneficial to Hancock’s
hypothesis that the Great Pyramid of Giza coded
precession. However, a preference for 26,000 over
25,920 was never discovered.
Hancock accurately states that one degree of
precession is completed roughly every 72 years. He
then associates precession to his chosen scalar by
pointing out that 43,200 divided by 600 is also 72.
Hancock likes the number 72 and claims that 72 is
often observed in ancient civilizations. And so, he
emphasizes the number 72, as if seeing 72 twice from
two different sources, regardless of how
unmeaningful those sources are, is self-evident that
something purposeful and special has been revealed.
Hancock also points out that 43,200 is the number of
minutes in 12 hours. Hancock’s tone in delivering
these revelations displays his personal fascination
that they must not only be purposeful, but indicative
of an advanced and mysterious knowledge. However,
the scientific method requires that it is demonstrated
that these relationships are not arbitrary, not
coincidental, not forced, and that they specifically
describe or imply precession using unbiased
measures the earth and the Great Pyramid.
For example, 25,920 itself is only somewhat arbitrary.
25,920 is, roughly, the number of years necessary to
complete one cycle (circle) of precession. If any
ancient culture clearly demonstrated the importance
of 25,920 (or 26,000) it could provide evidence for an
understanding of precession and beg for additional
investigation. So, if it can be shown that a scalar k =
25,920 was indicated by the dimensions of the Great
Pyramid, Egyptologists and historians would have
long agreed that the ancient Egyptians understood
and coded precession with impressive accuracy.
However, Hancock claims the scalar was 43,200 to
scale the Great Pyramid to the earth instead of 25,920
to scale it to precession.
Mathematical minded readers will notice that by
equating 43,200 to 25,920, Hancock has created an
additional degree of freedom with which he can look
for more numerical relationships. He now has two
numbers to find interesting associations with instead
of one. Yet he markets this freedom as another
unique and “discovered” truth. Connecting
Hancock’s dots for him, it needs to be shown that
43,200 is not arbitrary, or 43,200 is a product of non-
arbitrary factors, and that 43,200 or its factors relate
directly and profoundly to 25,920.
Hancock suggests the factors he uses are indeed
meaningful, though he does not specify exactly how
or why. In addition to (1 * 43,200), 43,200 has forty-
one other pairs of factors. Hancock especially likes
two of those forty-one pairs (600 * 72) and (3,600 *
12). Though 3,600 is notably missing from 25,920. He
also like (360 * 72) which are found in 25,920. He
highlights these various factors as if it is self-evident
that they are purposeful but does not offer any
specific evidence how or why they must be special.
Many do include the factor of 60, and Hancock favors
60 because of its obvious association to present day
time keeping and a 360-degree circle.
It is known that the Sumerians used a base 60
counting system about the same time that the
pyramids were built (Ifrah 2000), though solid
evidence is lacking that ancient Egyptians adopted a
base 60 system for either timekeeping or a circle
divided into 6 times 60-degrees. Regardless, clear
evidence and intention needs to be shown that the
factors 12, 72, 600, and 3,600 purposely connect
43,200 to 25,920 beyond simply explaining that the
selected factors are sometimes useful in other
applications. For instance, Hancock points out that
25,920 divided by 72 is 360 but even if it is certain that
the ancient Egyptians divided a circle into 360
degrees, the question remains why the number
25,920 uniquely indicates the number 43,200 simply
because 43,200 also shares factors of 72 and 360. To
date, the only thing available that relates precession
(25,920) to Hancock’s scalar (43,200) are a handful of
common factors which would be common for highly
composite numbers and highly factorable numbers.
Hancock provided another relationship; that the
number of seconds in 12 hours is 43,200, but this has
little use in an evidentiary sense. Not only must it be

8. 8
demonstrated that the ancient Egyptians divided a
day into 24 hours of 60 minutes, and 60 seconds, but
43,200 still fails to directly relate to 25,920 years of
precession. Hancock fails to provide any connection
for these relationships, and an objective inquiry into
the facts does not help. For Hancock, he simply rests
on pointing out that he appreciates large numbers
that possess factors found in present day time
keeping.
Anyone could continue this game with the other pairs
of factors. Frankly, it is surprising that Hancock
doesn’t. All the pairs of integer factors of 43,200 and
25,920 are included in Appendix D and certainly
people could study them and assign other
relationships various factors. Also shown in the
appendix is that each factor relationship emphasized
by Hancock are shared by many other large numbers
in addition to 25,920 and 43,200.
Conclusion
Graham Hancock’s assertion that the Great Pyramid’s
dimensions reveal knowledge of earth’s dimensions
certainly lacks proof but also fails to hold up to any
scrutiny as a viable theory. Each step argued by
Hancock; that the Great Pyramid was built to a
specific scalar, that the value of the scalar can be
definitively demonstrated, that the scalar indicates
precession, that the only way ancient Egyptians could
estimate precession is to leverage or borrow
knowledge from earlier advanced and unrecognized
civilizations, are each filled with flaws when taken
individually, let alone when strung together to
complete his narrative.
Objective evaluations of both the height and base
perimeter of the Great Pyramid show that neither are
closely comparable to any of earth’s radii when
multiplied by his claimed scalar of 43,200. AXIOM 3
states that the deteriorated condition of the pyramid
prohibits the calculation of a unique solution
equating the Great Pyramid to the earth.
Additionally, AXIOM 4 states that highly precise
decimal solutions on the order regularly claimed by
Hancock et al., cannot exist if the original pyramid
dimensions and the scalar were integers. Hancock
also claims that his scalar indicates a precise
knowledge of precession but offers no support other
than pointing out that the scalar and his selected
estimate of precession share some identical factors.
Although 43,200 and 25,920 share many factors,
Hancock selects only a couple that he believes are
profoundly important. He offers no evidence for their
unique importance other than stating that they relate
either to present day timekeeping or a 360-degree
circle.
Hancock et al. also like to speak about π being
precisely coded into the Great Pyramid. Proof for
such claims are also dubious because of the
deterioration of the pyramid and the natural
characteristic that π exists even when it is not
understood. AXIOM 1 states that π would be
incidentally incorporated into any pyramid built to
match the geometry of a circle. But even more
problematic for Hancock is that a pyramid cannot
simultaneously contain π and be proportional to two
different circles. Hancock claims the Great Pyramid is
proportional to both the polar radius of the earth
(Circle A), and the equatorial circumference (Circle
B). AXIOM 2 states that a comparison to two
distinctly different circles is mathematically
impossible if the ratio of the dimensions used in the
comparisons specifically define π.
While it is acknowledged that the ancient Egyptians
may have understood precession, support for such
knowledge cannot be found in the basic geometry of
the Great Pyramid. Additionally, the general rationale
that ancient knowledge of precession is noteworthy
because an enormously long observational period is
required to estimate it was contradicted in Appendix
E.
Hancock et al. continue to manufacture mystical
relationships and codes with respect to the pyramids.
Their intention is part of a broader mission to gather
support for a lost Pleistocene civilization that they
allege spread advanced knowledge throughout the
world before being destroyed by a comet about 12,800
years ago. These nonconformists function to revise
and reinterpret archaeological evidence and sites
(and sites that are entirely natural) world-wide to
narrowly and falsely formulate support for the lost
civilization. Their work is rarely scholarly, choosing
instead to selectively exploit mainstream archaeology
to the extent opportunities exist to distort or
repurpose evidence, and reject all archaeological
evidence and academic interpretations otherwise.
With respect to the pyramids, documenting and
debunking each new claim of mystical hidden
meaning is never ending, but it is amply clear that
their methods consistently rely on selecting input
values that confirm desired outcomes, as well as
misunderstood mathematical principles.

9. 9
APPENDIX A
RADII OF THE EARTH
It is impossible to singularly describe the radius of the
earth. This is primarily due to the earth being an
oblate spheroid instead of a sphere. In an oblate
spheroid, every major cross-section (cross-sections
that cut though the center of the earth) is an ellipse,
of varying sizes, except for the single circle formed by
the major cross-section at the earth’s equator.
Essentially, the earth is squashed at the poles and
bulges at the equator. So, the actual radius at the
poles is shorter than the radius at the equator. In
addition to this, the varied curvatures at the surface
of the earth equate to implied radii that are typically
not equal to the actual radii.
Radius of Curvature: Since the earth is squashed at
the poles the surface curvature of the earth at the
poles complements a sphere that is larger than the
actual size of the earth. The radii of curvature are thus
the radii that equate to spheres implied by the surface
curvatures. As a result, the radii of curvature at the
poles is greater than the radii of curvature at the
equator. Since the major cross section of the earth at
the equator is an actual circle (excepting for topical
geography), the East to West curvature exactly
matches the cross-sectional circle and the radius of
curvature and the actual radius are equal. This is
shown in Figure 2 below, where R is the actual radius
and N is the radius of curvature in the East/West
direction. At 0 degrees, the latitude at the equator,
both R and N are equal to 6,378 km.
Figure 2 shows three distinctly different methods for
measuring the radii of the earth and how each vary by
latitude:
Figure 2
The equator is at 0 degrees and the poles are at 90 degrees
M – The meridional radius of curvature. M describes
radii implied by the surface curvature along the
meridians. The meridians extend North/South.
N – The prime vertical radius of curvature. N
describes radii implied by the surface curvature
perpendicular to the North/South curves of M.
R – The actual measured radius from the center of the
earth to the surface.
These radii reflect both actual radii (R) and radii of
curvature (M and N). The actual radii are somewhat
intuitive. They are the actual distances from any point
on the surface of the earth to the center of the earth.
Since the earth is squashed (from top to bottom) and
bulges at the equator, these real distances from the
surface of the earth to its center are shortest at the
poles and greatest at the equator. At 90 degrees, the
actual radius, R, is only about 6,357 km, while at 0
degrees the actual radius is about 6,378 km.
The radius of curvature is the implied radius based on
the curvature at any point on the surface. At the
equator, because of the lack of squashing seen at the
poles, the curvature is tighter and equates to a smaller

10. 10
radius than at the poles. At 0 degrees, the implied
radius based on curvature, M, is only 6,335 km, while
at the poles (90 degrees), where the earth is squashed
flatter and the curvature is wide, a radius of 6,400 km
is implied.
Similarly, curve N represents the change in curvature
from the equator to the poles along the prime vertical
radius and represents curves perpendicular to M. N is
the most difficult cross section to visualize. Prime
vertical curves, N, are drawn horizontally from “C” to
“D” in Figure 3 below where meridional curves, M,
are drawn vertically from “A” to “B”.
Figure 3

11. 11
APPENDIX B
VOLUME OF THE GREAT PYRAMID
(Illustration Calculation)
This calculation illustrates how easy it is to equate the
Great Pyramid to the earth. Hancock et al. did not
perform this calculation in any published work
encountered.
Given that a pyramid is built with a perimeter equal
to the circumference of a specified circle, and the
height of that pyramid is equal that circle’s radius, the
following results would follow with respect to
volume:
• Eq. 9: Volume of a Sphere:
(4/3 * pi * radius cubed) = 4/3 πr³
• Eq. 10: Volume of a Pyramid:
(base length * base width * height) / 3 = (lwh)/3
For a pyramid with equal base sides (a square base):
• Eq. 11: Volume of a Pyramid = (base * base * h)/3
= (b² * h)/3
Rewrite the volume of a Pyramid in terms of a sphere.
Pyramid Base = (from Eq. 6 above), (2πh)/4 = (πh)/2
substitute the base shown in Eq. 6 into Eq. 11 gives:
• Eq. 12: Volume of a Pyramid = [((πh)/2)² *h]/3 =
(π² * h³)/12
Finally, dividing Eq. 9 (volume of a sphere) by Eq. 12
(volume of a pyramid) gives the ratio of a pyramid’s
volume to a sphere’s volume:
Eq. 13: Ratio of a Pyramid to a Sphere (in volume) =
(4/3 πh³) / (π² * h³)/12 = 16/π
So, to convert the Great Pyramid’s volume to the
volume of the earth (if the earth was a perfect sphere),
multiple the volume of the pyramid by 16/π and a
scalar factor (the scalar is cubed to account for
volume). Remember that Hancock claims that the
scalar is exactly 43,200.
• Continuing with Eq. 10: Volume of the Great
Pyramid of height 146.58 m and a base side of
230.34 m = 230.34² * 146.58 /3 = approximately
2,592,341 cubic meters
• Continuing with Eq. 12: Volume of the Great
Pyramid of height 146.58 m = (π² * 146.58³)/12 =
2,590,256 million cubic meters
Starting with the volume of the earth:
• Eq. 14: Volume of the Earth = 1.08321 * 10²¹ cubic
meters (exact modern measurement)
• Eq. 15: Volume of the Earth divided by Eq. 13 *
43,200 cubed: 1.08321 * 10²¹ cubic meters / (16/π *
43,200³) = 2,638,100 million cubic meters
Each of these calculations are close given the
parameters used. But it is easy to force them all to be
exact. If 22/7 is used in Eq. 12 instead of π, 2,592,341
cubic meters is the result. And if the volume of the
earth that equates to the radius provided above in Eq.
7 (6,332.26 km) is used, instead of the actual
measurement shown in Eq. 14, 2,592,341 cubic
meters is again the result.
Algebraically, these equivalent results are already
expected and understood for calculations with
underlying parameters that are also equal. Of course,
instead of pointing out these algebraic identities,
each calculation could be performed such that they
are all equal and the results presented as another
fascinating revelation.

12. 12
APPENDIX C
INDEPENDENT ESTIMATES BASED ON EVIDENCE
There was no intent in this analysis to discount the
knowledge or the ingenuity of the ancient Egyptians
or even their possible desire to use the Great Pyramid
to describe the size or movements of the earth. This
Appendix therefore offers the most likely
measurements of the Great Pyramid and how close
those measurements relate to the earth based on
objective considerations.
Great Pyramid Measurements
The accepted height and base measurements for the
Great Pyramid, in antiquity are (Levy 2007):
• Height: 280 cubits
• Base side: 440 cubits
• Perimeter: 1,760 cubits
If these measurements are correct, they immediately
reveal that the pyramid was constructed with the
geometry of a circle in mind and with the value of π
being approximated by the natural number 22/7.
Substituting 22/7 for π, and setting the radius to 280
cubits, a circle is produced with a circumference of
1,760 cubits. This is equal to the base perimeter of the
Great Pyramid. These measurements seem
harmonious with each other, if merely because each
are whole integers. Of course, it is possible that the
ancient Egyptians understood π more precisely than
22/7 and that one or more of these preferred
measures are incorrect or not integers.
For example, if the ancient Egyptians understood π
perfectly, and they designed a pyramid with a base
perimeter of 1,760 cubits, the resulting height would
be 280.11 cubits instead of 280.00 cubits exactly. If the
height was selected as 280 cubits exactly, then the
implied base perimeter would be 1,759.20 cubits
instead of 1,760.00 cubits.
The use of round numbers seems more harmonious
and more likely. Also 22/7 is an extremely close
approximation for π. In fact, 22/7 is so close that if the
ancient Egyptians laid out a circle with a radius of 100
cubits (about 172 feet), the difference between the
actual circumference to the calculated circumference
using 22/7 is only 0.253 cubits, or 5.2 inches of a circle
about 1,079 feet in circumference. Such an error
would be very difficult to confirm performing
practical experiments and measuring the resulting
circumference and radius explicitly.
Scalar Measurement
Rejecting Hancock’s value of k = 43,200 as a given
starting point, unbiased calculations of the scalar
based on the known size of the earth and the accepted
size of the Great Pyramid are now produced.
Note that the complete set of radii, from Appendix A,
are in the range of about 6,335 km to 6,400 km.
Converting this to cubits is 12,101 cu to 12,225 cu.
Using the accepted size of the Great Pyramid, height
and perimeter of 280 cu and 1,760 cu, respectively, k
is calculated and expressed as a range since the
possible radii of the earth is a range.
• Earth’s radii range: (12,101.2, 12,225.4) in cubits
• Earth’s circumference range: (76,064,686,
76,845,371) in cubits
• Range of k implied by the pyramid’s accepted
height: (43,219, 43,662)
• Range of k implied by the pyramid’s accepted
perimeter: (43,219, 43,662)
As has been demonstrated and discussed previously,
it is already clear that the implied range of “k” for both
the height and perimeter would be equal since both
measurements are compatible with a 22/7 circle.
Starting with a proper range of radii, and the accepted
measurements of the Great Pyramid, the scalar 43,200
isn’t even in the range of expected possibilities.

13. 13
APPENDIX D
FACTORS of 43,200 and 25,920
Listed in this appendix are the pairs of factors for both
43,200 and 25,920. One could play with these at
length to find interesting ways to relate them to each
other, the Great Pyramid, or some other number
associated with the earth or ancient civilizations. The
numbers in RED BOLD highlight matches between
43,200 and 25,920.
43,200 factors: (43,200, 1), (21,200, 2),
(14,400, 3), (10,800, 4), (8,640, 5), (7,200, 6),
(5,400, 8), (4,800, 9), (4,320, 10), (3,600, 12),
(2,880, 15), (2,700, 16), (2,400, 18), (2,160,
20), (1,800, 24), (1,728, 25), (1,600, 27), (1,440,
30), (1,350, 32), (1,200, 36), (1,080, 40), (960,
45), (900, 48), (864, 50), (800, 54), (720, 60),
(675, 64), (600, 72), (576, 75), (540, 80), (480,
90), (450, 96), (432, 100), (400, 108), (360,
120), (320, 135), (300, 144), (288, 150), (270,
160), (240, 180), (225, 192), and (216, 200).
25,920 factors: (25,920, 1), (12,960, 2), (8,640,
3), (6,480, 4), (5,184, 5), (4,320, 6), (3,240, 8),
(2,880, 9), (2,592, 10), (2,160, 12), (1,728, 15),
(1,620, 16), (1,440, 18), (1,296, 20), (1,080, 24),
(1,037, 25), (960, 27), (864, 30), (810, 32),
(720, 36), (648, 40), (576, 45), (540, 48), (518,
50), (480, 54), (432, 60), (405, 64), (360, 72),
(346, 75), (324, 80), (320, 81), (288, 90), (270,
96), (240, 108), (216, 120), (192, 135), (180,
144), and (162, 160).
Someone might consider that the sheer number of
matching factors connotes a meaningful relationship.
While a meaningful relationship between 25,920 and
43,200 can never be absolutely disproven, it is
certainly possible to show that meaning does not
exist simply because numerous factors are shared – or
even factors important to Hancock are shared. To
demonstrate this, many unrelated numbers are
selected below to highlighted how easily one number
might share factors with another.
Essentially, any large number will work well so long
as it includes factors for most of the integers from 1
through 10 (noting that the integers 4, 6, 8, and 9 can
be further factored), though 7 is notably absent from
both 43,200 and 25,920. Finding such numbers is
relatively easy by simply grabbing a handful of
integers between 1 and x (where x can be as small as
5) and then multiply them together. Such numbers
can be useful in various applications and are similar
to highly composite numbers. These numbers can be
factored into many base integers and all the integers
produced by multiplying combinations of the base
integers. So, assessing the relationship of such a
number to nature or other numbers will usually
provide many options to investigate and shape stories
around.
The ancient Egyptians well understood multiplying
and factoring, and certainly could have preferred
large highly factorable numbers. But if so, such
numbers will relate to each other and to other
numbers in multiple ways, intentionally or otherwise.
Because of this, common factors alone can scarcely
provide support for how two numbers can be
partnered together.
For comparison, here is a list of many numbers that
satisfy the property of sharing factors with 25,920.
Each are produced by choosing multiple small
integers and multiplying them together.
Each contain the following integers that Hancock
particularly likes: 12, 60, 72, and 360.
25,920 = 2⁶ * 3⁴ * 5
43,200 = 2⁶ * 3³ * 5²
23,040 = 2⁹ * 3² * 5
25,200 = 2⁴ * 3² * 5² * 7
28,880 = 2⁷ * 3² * 5²
30,240 = 2⁵ * 3³ * 5 * 7
32,400 = 2⁴ * 3⁴ * 5²
34,560 = 2⁸ * 3³ * 5
38,800 = 2⁵ * 3⁵ * 5
40,320 = 2⁷ * 3² * 5 * 7
45,360 = 2⁴ * 3⁴ * 5 * 7

14. 14
APPENDIX E
INVESTIGAGTING THE PRECISION OF PRECESSION
(Bonus Illustration using the Mayan Calendar)
The value used by Hancock et al., 25,920, is close to
the actual value of precession. Today precession is
measured to be about 25,771.5 years. So then, to show
that the Great Pyramid does not code precession, it is
technically insufficient to simply show that 25,920 is
not coded. On one hand, it is fair for Hancock to
target 25,920; if the ancient Egyptian did understand
precession, perhaps they did estimate it to be 25,920
since a more precise measurement could have been
beyond their capabilities. On the other hand, it is
curious why Hancock claims that the Great Pyramid
was built using hyper-advanced knowledge that
humans can’t duplicate today, while touting a value
for precession that is not nearly as precise as known
today.
In any case, the likelihood that the Great Pyramid
incorporated precession into its dimensions requires
more investigation to fully understand. To begin
with, readers are reminded that nothing near 26,000
was claimed as a Great Pyramid scalar to begin with.
Rather, Hancock connects 43,200 to the Great
Pyramid, and only connects 25,920 via a few factors
with 43,200. It can also be noted that, in Appendix C,
the actual scalar was calculated in the range (43,219,
43,662), rejecting Hancock et al. claims that the scalar
is precisely 43,200 years. In this analysis estimates are
usually expressed as a range, since the earth is not a
perfect sphere and has a range of radii.
Hancock suggests the scalar is 43,200. His inspiration
and confidence is derived from the simple
observation that various factors are shared by both
43,200 and 25,920. Appendix D lists the 42 sets of
factors that the 43,200 can be factored into. The
supposition that meaning between 25,920 and 43,200
is present simply because some of their factors are
similar can’t be entirely discounted though strong
skepticism persists when noting that many other
ignored factors exist between the two numbers, and
the factors that Hancock selected are regularly seen
in other numbers.
It is also fair to restrict considerations to integers, as
Hancock tends to do. Integers tend to be harmonious
with nature and easily conveyable interpretations of
the observed world, though it is not strictly necessary.
If calculations were expanded to the set of all real
numbers, then there would be infinitely more
flexibility to associate one calculation or observation
with another.
However, if consideration is granted that the Great
Pyramid coded precession or earth’s dimensions to
decimal accuracy (perhaps ever more accurately than
25,920), then it must be conceded that the Great
Pyramid was not built to round integer dimensions in
the first place (280 cubits height, and 440 cubits base
side are the accepted measures) unless the radii of the
earth and precession are both coincidentally
measured as perfect round integers in cubits. Put
another way, if the ancient Egyptians precisely
calculated precession or earth’s dimension to decimal
accuracy then they would be forced to build a
pyramid that couldn’t be expressed with round
integers excepting for the infinitesimally small
chance that the earth’s dimensions are exact integers
when measured in cubits, or by choosing a scalar that
is not an exact integer.
That might sound acceptable in principal, but the
problem for Hancock et al. is that if the pyramid was
not exactly 280 cubits in height and 440 cubits in
length for a base side, if it was instead some decimal
measure within the range of reasonable possibilities,
it could never be known what the original dimensions
were because of the deterioration that has taken
place.
AXIOM 3: A single unique solution equating
the Great Pyramid to the dimensions of the
earth does not exist because the deteriorated
state of the Great Pyramid does not uniquely
indicate uniquely provable dimensions.
Likewise, if the scalar is not a nice integer that can be
factored into many other integers, then nothing in
Hancock’s hypothesis makes any sense at all.
Hancock et al. must live by the integer or die by the
integer. This contradiction is at odds with Hancock et
al. regularly reporting measures of the pyramid and
the earth to many decimal places of precision.
AXIOM 4: A high precision solution
equating the Great Pyramid to the earth does
not exist if they pyramid was built to integer
dimensions and multiplied by an integer
scalar.

15. 15
One either assumes the Great Pyramid was built to
harmonious integer measures and proceeds with a
search for harmonious integer relationships, or one
must immediately give up any hope of demonstrating
a singularly profound and hidden relationship given
the ambiguity in determining an unknown decimal
measure contained in the pyramids original size.
Hancock et al. seem to prefer integer relationships in
their initial introductions, but then contradict
AXIOM’s 3 and 4 as they find ways to force results to
high decimal place accuracy. In this way they can
express their mystical arguments that the ancient
Egyptians coded knowledge of precession (in
addition to various earth’s radii), in a way that
resonates with others. If they attempted to show that
the scalar is, for example, 43,413.78 (or any other value
within the range given in Appendix C), it would be
very difficult to excite the layperson about how
43,413.78 relates to 25,920 or the precise
measurement of 25,771.5. But then deeper into their
presentations they force certain results and report
that various obscure calculations matched to many
decimal places of accuracy.
Why is a careful precession estimate
considered so advanced or mystical in
the first place?
Here is more about the Precession of the Equinoxes,
as explained by Graham Hancock (Hancock 2017).
Hancock et al. gravitate towards things like
precession because estimating precession is
indicative of a reasonably advanced civilization. This
is because a reasonably long period of careful
measurements is required to first observe precession,
and then to calculate its cycle with some precision.
Associating the knowledge of precession with
advanced astronomical observation and record
keeping techniques is not disputed. Indeed, it takes
roughly 72 years for the earth to complete one degree
(of 360) of precession. It seems reasonable that it
might take 50-100 years for any civilization to
generally notice precession and then begin to
calculate it. However, the enormously long time that
most pseudoscientists claim is required to explicitly
calculate long period phenomena is disputed.
The observational requirements to estimate long
period phenomena (in the absence of advanced and
modern scientific equipment), is a major cause for
pseudoarchaeologists to gravitate toward beliefs in
older hyper-advanced civilizations. This is a
fundamental flaw in their logic because refining
estimates for long period phenomena can be
accomplished in a fraction of the time the
phenomenon completes one cycle. It is not necessary
to observe the phenomenon start to finish to
calculate its duration with some accuracy, as the
pseudoscientists’ arguments tend to require.
Illustration – The Length of the Mayan
Year
To first illustrate this point, consider the length of the
Mayan year. The exact length of an earth year is
365.24219 days, or 365 days, 5 hours, 48 minutes, and
45 seconds (“Tropical Year.” 2019). This is very close
to the length of the Mayan year of 365.24204 days. To
refine the length of a year to decimal accuracy,
pseudoarchaeologists like to claim that centuries of
observation are required. They claim that it would
naturally take four years to observe that a year is
roughly 365.25 days long.
This rationale progresses as follows: Begin building a
calendar and count the days in a year. Only after four
years will ancient astronomers be capable of saying
that their calendar, relative to the observation of the
stars, is off by about one day. Following this linear
observation method, the ancient astronomers update
their calendar from a year is equal to 365 days, to a
year is equal to 365.25 days. (they observed 1,461 days
passing until the sun rose or set over the same
location, instead of 1,460 days). Continuing in this
fashion, they would need another 100 years to notice
that they are again off by day. This time they can
refine the length of a year from 365.25 days to 365.24
days by dividing 36,524 days by 100 full periods of the
sun rising or setting over the same spot (Razzeto
2009).
This method is certainly reliable and might be
generally necessary for anyone unable to interpolate
between days or years, thereby making partial period
estimates using a secondary timekeeping source or
intermediate measurements. But, secondary
timekeeping methods, intermediate measurements,
and even numerical calculation methods are available
to better refine an estimate. Even if the ancient
astronomers fail to employ an independent and
secondary method to accurately measure a portion of
a day, at the end 365 days they will still notice that the
final day concluded about 1/2, or 1/3rd
, or 1/4th
a day
too soon in comparison to the movement of stars.

16. 16
Such an observation could prompt the development
of methods to suppose when the sun once again sets
or rises “on time” without waiting 1,461 days to
explicitly see it.
Based on the Mayan calendar it is believed that the
Mayan civilization calculated the length of a year as
365.24204 days (Douma 2008). However, other
sources convolute the matter and cite a much more
accurate measurement of 365.2422 days (Razzeto
2009). Naturally, these seemingly more accurate
estimates are more likely to be adopted by the pseudo
world because they best fit the fantastic pseudo
narrative.
Using the explicit method of relying on full
observational periods to adjust the length of a year
only after observations are off by one full day implies
that the Mayan year of 365.2422 days would require
approximately 5,000 consecutive years of
observations. For those who accept the explicit
method as a reasonable explanation, it does indeed
beg the question whether the Mayans received help
from an earlier advanced civilization or aliens.
Unfortunately, pseudoarchaeologists, who rely upon
such limited rationale, proceed immediately to the
debate between ancient unknown civilizations versus
alien visitation rather than seeking other
explanations.
But even if the most accurate estimate in the Mayan
year is considered, an explicit method of observing
for thousands of years is not required. The explicit
observation method ignores the potential use of
numerical methods such as period over period
averaging, plotting, extrapolation and convergence
algorithms, etc., for refining estimates and
extrapolating final estimates based on partial period
or intermediate measurements.
For example, if an ancient civilization’s astronomers
could measure a fraction of a day to within 10 minutes
(using the stars or some other time keeping method),
then after a single year they could calculate the length
of a year as 365 days plus 1/4th
of a day +/- 10 minutes.
That equates to a range estimate of (365.235, 365.249)
days, after only one single observational year. The
error in this method is the single error attributed to
the final portion of a day measured to within +/- 10
minutes.
If daily calculations are performed for 1,000 days, on
day 1,000 the total error is still limited to the final
error in calculating a fraction of day 1,000 (still
assumed to be about 10 minutes error). That error is
now distributed over 1,000 days instead of just 365,
and the overall year estimate becomes even more
refined. After 10 years not only is the error much
smaller in proportion to the observation period, but
thousands of day-over-day estimates that can be
analyzed with other numerical methods (averaged,
plotted, trended, etc.).
Modeled randomized observations with an average
day-over-day error rate of +/- 10 minutes shows that
the range can be narrowed to (365.2417, 365.2431)
after 3,652 days (about 10 years). The midpoint of this
range is 365.24226 and compares well to the actual
length of 365.24219 days, and most accurate Mayan
estimate of 365.2422 days. Running the model again
but with an average day-over-day error rate of +/- 2
minutes, produces a range of (365.24209, 365.24237)
and midpoint estimate of 365.24223 days. Remember
that it is important in this method to be able to
measure a portion of a day using the stars or some
other manufactured method or apparatus.
Using a star to mark the onset of an event is not a
difficult task. This can be demonstrated by the 18th
century English clockmaker John Harrison, winner of
the Board of Longitude’s longitude prize. Before
winning that prestigious prize, Harrison regularly set
his handmade wooden clocks to within 2 seconds by
noting a star disappearing behind his neighbor’s
chimney (Quill 1963). The ability to note the onset of
such an event in time is not difficult. Rather,
accurately measuring the amount of time between
two known events is where some ingenuity is
required. This can be illustrated by using the distance
between two points on a line rather than considering
time. When two marks are drawn on the earth some
distance apart, even if the placement of those two
marks are well known, it might be difficult to specify
the exact distance between the two. In this way the
Mayan’s would need some method for estimating the
passing of portions of a single day. Numerous such
ideas could be discussed and could have occurred.
The table in Figure 4 demonstrates how the modeled
results got tighter by increasing the observation
period from 20 days to 10 years. These results also use
some simply numerical methods such as averaging
and iterated interpolated methods. Such simple
numerical methods are at least as old as Archimedes
(ca. 200 B.C.) (Burden & Fairs 1989). More
complicated methods could also be considered, both
observationally and numerical, that could allow for
even faster convergence.

17. 17
The following table displays the modeled results
using randomly generated daily measurements. The
first column assumes daily measurements within +/-
10 minutes accuracy, and the second column to
within +/- 2 minutes. After the random observations
the numerical methods are applied to estimate a
range around the expected actual length of a year
measured in days. The table is an example of how
quickly estimates can converge when using a
secondary time estimate and some careful application
of numerical methods:
Figure 4
Actual Year: 365.24219 days. Mayan Year: 365.24204 days.
Back to Precession
Applying this same type of ingenuity and
observational techniques to an ancient civilization’s
search for precession could mean that ancients may
have enhanced their estimates using fractional
observations. Also, as the number of measurements
grows, the use of numerical methods makes it
possible to estimate a range even more precisely.
For example, if the ancient Egyptians could discern
precession through the movement of a rising or
setting star after the star moves 1/7th
of a degree, they
would discern the 1/7th
degree movement after
approximately 10 years (10.15 years = 1/7th
of a degree).
And measuring the location of a star to 1/7th
of a
degree might not be as difficult as it might initially
seem. For example, if ancient astronomers used a
large flat area for making astronomical observations
and laid out a method for marking locations at one-
mile distance from the observation location, then two
points separated by approximately 92 feet would
represent a single degree. And 1/7th
of a degree is
represented by approximately +/- 13 feet. It is not too
difficult to imagine a group of ancient astronomers
siting at the observation location and noting the
location of a bright star rising or setting to within 13
feet of its actual location (from one mile). Given that
they could complete these observations over the
course of many days they could settle on a reasonable
estimate. Of course, precession itself would slightly
alter those day to day observations but by only 1.3
inches over a 30-day period. Such an error could
either be ignored or accounted for using the latest
overall estimate of precession.
Imagine that these ancient astronomers begin taking
measurements with a precision of 1/7th
a degree (or 13
feet from one mile). Suppose that their very first
measurement indicated movement of 1/5th
of a degree
after a duration of time of 15 years (14.21 years is
exact). But, of course, their estimate only carries a
precision itself of 1/7th
degree. Their point estimate
would then be 27,000 years (360 degrees divided by
1/5th
degree times 15 years). But this early estimate is
not very precise since it includes an observational
error of +/- 1/7th
a degree on an observation of only
1/5th
degree. In fact, the measurement of 1/5th
degree
could have been much larger or smaller given the
assumption that the measurement error is 1/7th
degree. Regardless, the error component is large in
comparison to the observation and the following
calculation for the first observation illustrates the
range of accuracy.
Observation 1 (15 years):
• Lower bound
= 1/5 degree + 1/7 degree = 12/35 degree
= 360 * 35/12 * 15 years = 15,750 years.
• Upper bound
= 1/5 degree – 1/7 degree = 2/35 degree
= 360 * 35/2 * 15 years = 94,500 years.

18. 18
(15,750, 94,500) years is quite a wide range after one
observation and calculation.
But successive measurements greatly increase the
p0ssibility for making more accurate measurements
because the successive measurements still have only
the single observational error of 1/7th
degree and
eventually they can be subjected to numerical
methods. If a second observation estimates that 2/5th
a degree is completed after 27 years (28.41 years is
exact), then the error, which is random but still
systematically equal to +/- 1/7th
degree, is now half the
size in comparison to the overall measurement being
taken. This is good though our ancient astronomers
had to invest twice the time to achieve it.
The point estimate after the second observation at 27
years would then be 24,300 years (360 degrees divided
by 2/5th
degree times 27 years). But still with a range
of +/- 1/7th
a degree.
Observation 2 (27 years):
• Lower bound
= 2/5 degree + 1/7 degree = 19/35 degree
= 360 * 35/19 * 27 years = 17,905 years.
• Upper bound
= 2/5 degree – 1/7 degree = 9/35 degree
= 360 * 35/9 * 27 years = 37,800 years.
Note that the range, (17,905, 37,800), has narrowed
significantly. Continuing:
Observation 3 (44 years):
• Point estimate = 360 * 5/3 * 44 years = 26,400
• Lower bound
= 3/5 degree + 1/7 degree = 26/35 degree
= 360 * 35/26 * 44 years = 21,323 years.
• Upper bound
= 3/5 degree – 1/7 degree = 16/35 degree
= 360 * 35/16 * 44 years = 34,650 years.
The range, (21,323, 34,650), is even more narrow
because the error of +/- 1/7th
degree is now getting
much smaller in comparison to the observation of
3/5th
degree.
Note that other annual observations are possible as
well. The ancient astronomers might discern that
their star moved 1/2 degree after 34 years (35.51 years
is exact). The error in that observation is still roughly
+/- 1/7th
degree (roughly: because each error of each
observation is random though assumed to have an
average error of about 1/7th
degree). The range given
at 34 years would be (19,040, 34,272). Having multiple
estimates in this manner allows for even further
refinement of the estimate by applying numerical
methods to the complete set of observations.
Using mathematics along with careful observation
techniques, it is not difficult to imagine ancient
astronomers noticing and then making initial
(though perhaps rudimentary) estimates of
precessions after a few decades of observational
recordkeeping.
At any rate, ancient Egyptians certainly wouldn’t
require 1,000’s of years of observations to notice that
stars were slowing rising and setting in different
locations, nor 25,920 years to estimate the time it
would take to complete a full cycle. Of course, if they
did, they would have observed that precession
completed after only 25,771.5 years.
Model Results for Precession
Observational results were modeled assuming
observational errors are random and normally
distributed with a mean of 1/7th
degree. Then simple
numerical methods were used to analyze the set of
modeled results which produced a point estimate of
about 25,485 years after 100 years of observations.
Decreasing the precision of the observation from
1/7th a degree to only 1/2 a degree produced a point
estimate of 26,884 after 100 years. It should be noted
that the modeled results use only a single set of
randomized observations – anecdotal to what a group
of astronomers might experience when performing
real observations. The model results do not display
the expected results if the model was run thousands
of times. The model also converges to the actual value
of 25,771.5 years rather than 25,920 years.
Figure 5 displays the model results. Actual results
were recorded annually but the table has been
summarized for the years shown.

19. 19
Figure 5
It is not perfectly clear why an ancient civilization
might settle on 25,920, other than acknowledging
some of the harmonious aspects that number has
with whole integer multiplication. Of course, it might
also be that that ancient Egyptians did observe
precession but estimated precession to some
duration other than 25,920 years.
It is also not clear whether the ancient Egyptians
could have measured precession with high precision
by the time they designed and constructed the Great
Pyramid of Giza, but it is not altogether impossible
given the history, accomplishments of earlier
dynasties and other earlier civilizations, and the table
above which shows that even at 1/2 degree
observational accuracy, the value of 25,920 could
have been a debatable option within 100 years of
observations.
Regardless, to the extent that any ancient civilization
borrowed observations and measurements from
earlier astronomers, or relied entirely on their own, it
can be shown that precession can be estimated in
ancient times without indicating the existence of any
extremely ancient and hyper-advanced civilizations.

20. 20
APPENDIX F
WHEREVER π EXISTS, Ø EXISTS
The pages above have already illustrated how a
pyramid could include a precise value for π in its
geometry entirely by accident. Alternatively, the
approximation of 22/7 may have been purposefully
used as a proxy for π during the construction of the
Great Pyramid. It was also shown that due to the
deterioration in the Great Pyramid, if a value for π was
purposefully used, and that value was more accurate
than 22/7, it would be impossible to determine what
the actual value was.
Whether purposeful or accidental, it has been shown
that discovering π in the geometry of the Great
Pyramid does not indicate a precise knowledge of π
as a transcendental number, no matter how accurate
one supposes the discovered value of π to be.
Likewise, a close approximation for to Ø would also
be discovered in the Great Pyramid as a natural result
of either using 22/7 directly or accidentally building
in a precise value for π. In other words, wherever π is
seen, close approximations for Ø are also present. And
as a matter of fact, Ø is even more accurately
approximated when 22/7 is used as a proxy for π. This
is true because the square of four divided by π and the
and the square root of 5/6 π, are both close
approximations for Ø.
The mathematical formula for this approximation is
simply:
• Eq. 16a: (4/π)² ≈ Ø
• Eq. 16b: �
5
6
∗ 𝜋𝜋 ≈ Ø
Where Ø is the Golden Ratio.
• Eq. 17: Ø = (1 + √5)/ 2
So then, Eq. 16a ≈ Eq. 17
(4/π)² ≈ (1 + √5) /2
( 4 /3.14159…)² ≈ (1 + 2.23607…) /2
(1.27234…)² ≈ (3.23607…) /2
1.621138938… ≈ 1.618033989…
This approximation differs from the precise value
of Ø by 0.1919%.
And, Eq. 16b ≈ Eq. 17
�
5
6
∗ 𝜋𝜋 ≈ (1 + √5) /2
√2.6179939 ≈ 1.618033989…
1.618021594… ≈ 1.618033989…
This approximation if extremely close and only
differs from the precise value of Ø by 0.0008%!
For Eq. 16a, when using 22/7 for π, the
approximation is even closer:
(4 / 22/7)² ≈ (1 + √5) /2
( 4*7 / 22)² ≈ (1 + 2.23607…) /2
(1.272727…)² ≈ (3.23607…) /2
1.619834711… ≈ 1.618033989…
This time the approximation only differs by
0.1113%
Knowing that whenever π is found Ø must follow,
some of the equations previously introduced can be
rearranged, some algebra applied, and Ø will be
directly found in the Great Pyramid.
Consider Eq. 5 from page 4:
• Eq. 5: Pyramid perimeter = 4*b = (2πh)
Where “h” is the pyramid’s height “h”, and the
base perimeter of that pyramid was set equal to
the circumference of a circle with radius “h”.
Rearranging this equation gives:
• Eq. 18: 1/π = h /2*b
Multiply both sides by 4 and squaring both sides
gives:
• Eq. 19a: (4/π)² = (2*h /b)²
The left side of Eq. 19a is nearly equal to Eq. 16a,
Ø. So then, it is shown that the square of twice
the Great Pyramid’s height divided by the length
of a base side is equal to Ø. Ø can be found in
other ratios too, but each time it must be a
natural occurrence if π can also be found because
Eq. 16a is simply and coincidentally a close
approximation for Ø. Similar relationships can be

21. 21
contrived to relate Eq. 16b to specific elements of
the Great Pyramid.
For instance, Ø is most commonly shown in the
Great Pyramid as the ratio of the sum of the
surface areas of each face, to the surface area of
the base.
Algebraically, this is:
• Eq. 20: 4*F /b² = Ø
Where “F” is the area of a face.
Using the diagram in Figure 6, “F” is calculated and
after some algebra and simplification of equations, it
can be shown how naturally Eq. 20 must equal or
approximate Ø for any pyramid that has embedded π.
And π will always be embedded when constructing a
pyramid with the geometry of a circle in mind.
• Eq. 21: F = (b*H) /2
Using the Pythagorean formula to solve for “H”:
• Eq. 22: h² + (b/2)² = H²
H = √[h² + (b/2)²]
Substituting Eq. 22 into Eq. 21 gives:
• Eq. 23: F = {b * √[h² + (b/2)²] }/2
Substituting Eq. 23 into Eq. 20 gives:
• Eq. 24: 4 * [ {b * √[h² + (b/2)²] }/2 ] /b² = Ø
Eq. 24 can be reduced to:
• Eq. 25: 2*√[h² + (b/2)²] /b = Ø
Square both sides and reduce:
• Eq. 26: 4*h²/b² + b²/b² = ز
(2*h/b)² + 1 = ز
From Eq. 18 substitute 1/π = h /2*b
• Eq. 27: (4/π)² + 1 = ز
From Eq. 16a substitute (4/π)² ≈ Ø
• Eq. 28: Ø + 1 = ز
Eq. 28 is a known identity for Ø and one of the
reasons is it so special.
So, beginning with Eq. 20 and continuing to Eq 28, it
has been shown that for any pyramid, if the height is
set equal to the radius of a circle, and the perimeter is
set equal to the circumference of that same circle,
then approximations for Ø will be found throughout
the pyramid if only because Ø is closely approximated
by (4/π)² and π is naturally present in such a pyramid.
Finding a close approximation of Ø in the geometry
of such a pyramid is always purely accidental.
Another identity that is commonly discovered is that
the area of each face, triangle (A, C, E), is equal to the
square of the height.
In the form of an equation:
• Eq. 29: F = h²
(b*H) /2 = h²
H = 2h² /b
2H /b = (2*h /b)²
Using Eq. 26 and Eq. 28:
2H /b = Ø
Figure 6

22. 22
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