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8/12/2019 Phi in the Great Pyramid
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Phi in the Great Pyramidhttp://www.sacred-geometry.es/en/content/phi-great-pyramid
1.- Introduction
The Great Pyramid of Giza, in Egypt, fascinates many of us. It is an astonishing construction
built with an incredible accuracy, far beyond the one achievable by our current technology. In
this article we discuss some important properties of its design which are closely related to
Sacred Geometry. We will show that the ratio of the apotem to half the base obeys the Golden
Ratio, and that the perimeter of its base equals that of a circle with a radius equal to its height.
We will also see that the angles of the Great Pyramid hide the Euler number. As a preparing we
need to review the properties of the Kepler triangle and the problem of the Squaring of the
Circle. Then we will show that the dimensions of the Earth and Moon obey the same proportions
as the dimensions of the Great Pyramid, and that they can be directlyobtained using the
Phythagorean 3-4-5 triangle. And finally we will show how the exact dimensions of the Great
Pyramid obey a very simple formula related to the cubes of some specific integer values.
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2.- Squaring the Circle
Consider a general right-angle triangle of base b, heighthand hypothenuse a. Its circumscribing
circle will have the same perimeter as a square of size hprovided that the sides of the triangle
satisfy the relationship (Figure 1a)
a=4hah=4
This is often known as the Squaring of the Circle. If the square is formed with a side equal
double the triangle base 2b, the same problem is usually stated in terms of the circle that has a
radius equal the triangle height h(Figure 1b). In this case, the squaring of the circle is achieved
if the height and the base are related as
2h=8bhb=4
(a) (b)
Figure 1:Two ways of squaring the circle starting from the same right-angle triangle.
It is obvious that the two relationships above cannot be satisfied simultaneusly, because given
two sides of a right-angle triangle the third one must obey Phythagoras theorem. However, if we
relax the above equalities to good approximations, we see that the sides of the triangle wouldhave to be related in a geometric progression:
ah4hb4h2=ab
A question arises naturally: is that possible? And in this case, which is the accuracy of the
squaring of the circle approximated that way?
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3.- Kepler triangle
We know there is only one right-angle triangle whose sides are in an arithmetic progression, the
Phythagorean 3-4-5 triangle. Similarly,there is only one right-angle triangle with edge lengths
in a geometric progression, known as Kepler triangle. Do you guess what magic number is
involved in this triangle? Yes, as it could not be another way, it is the Golden Ratio! The sides ofKepler triangle obey the progression 1::. The reader can check that this triangle approximates
the Squaring of the Circle with an error of 0.096% (Figure 2), which exemplifies the known
relationship between and :
4
Figure 2:The squaring of the circle in Kepler triangle.
In the following section we will show that the Great Pyramid squares the circle with a
better approximation than the one obtained from Kepler triangle.
4.- Sacred Geometry in the Great Pyramid
The following figure shows the definition of the apotem (a), the height (h) and half base (h) of a
typical square pyramid. At the right we list the accepted dimensions of the Great Pyramid in
three different units of measurement:
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2b H
440ec
280ec
755.9ft
481.0ft
230.4m
146.6m
Table 1:Accepted dimensions of the Great Pyramid of Giza (from Wikipedia), given in egyptiancubits (ec), in feet (ft) and in meters (m).
The first property that we want to highlight is the presence of the Golden Ratio in the
Great Pyramidwith a high accuracy, as the ratio of the apotem to half the base:
ab=2202+2802220=1.618590347r=0.03%
This ratio provides an approximation rto the Golden Ratio . As a consequence, the trianglea-
b-hshown in Table 1 is very close to a Kepler triangle. Therefore, the Great Pyramid also
exemplifies in stone the Squaring of the Circle!If you draw a circle with radius equal to
its height, its perimeter will be the same as the base of the pyramid with an error as low
as 0.04%:
square8b=1760 eccircle2h=1759.2919 ec
The fact that the Great Pyramid approximates the Squaring of the Circle with a better accuracy
than Kepler triangle is due to the use of the rational approximation of /2 in the ratio of height
to base (Figure 3):
2117=440280Therefore the rational approximationrof is given by:
r227=3.142857143=0.04%
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(a) Squaring of the circle in the Great Pyramid (b) Rational approximation of /2 in the height tobase ratio of the Great Pyramid.
Figure 3
Less known is the fact that the dimensions of the Great Pyramid also hide another important
number in science, the Euler number e. Rick Howard'sresearchhas shown that the ratio of the
two angles in the triangle a-b-hin the Great Pyramid (Table 1) provides a very accurate
approximation erof e:
=tan-1(280220)=tan-1(1411)=51.842773413
er
290-=2,717323980=e0.035%
The following table illustrates and summarizes the three key numbers contained in the
dimensions of Great Pyramid of Giza:
KeyNumber
ExactValue
GreatPyramid
Error
1.618033989 1.618590347 0.034%
3.141592653 3.142857153 0.040%
e 2.718281828 2.717323980 0.035%
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Table 2:Key numbers hidden in the dimensions of the Great Pyramid of Giza.
5.- The Great Pyramid contains the diameter of the Earthand Moon
If the preceding has not left the reader astonished, what follows probably will. The squaring of
the circle in the Great Pyramid of Giza as shown in Figure 3a is a diagram that allows the precise
determination of the diameter of both the Earth and the Moon. You "only" need to draw an
incribed circle inside the square (the Earth) and a small circle centered at the apex and tangent
with the preceding one (the Moon). At each side on top of them there are two exact 3-4-5
Pythagorean triangles (Figure 4).
Figure 4:The proportions in the Great Pyramid and the Phythagorean 3-4-5 trianglelead to the exact dimensions of the Earth and Moon (see also Figure 5).
This old known 3-4-5 triangle allows to calculate the "magic" number 720 as follows:
3+4+5=12345=60}1260=720
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From it and the proportions of the Great Pyramid as shown in Figure 4, the diameter of the
Earth and Moon in miles can be directly determined!!!
Earth:11720=7920 milesMoon:3720=2160 miles
6.- The architects of the Great Pyramid had a secretFollowing thisexcellent articleby Joseph Turbeville, in this section we're going to show how the
exact dimensions of the Great Pyramid obey a very simple formula related to the
cubes of some specific integer values. Those values can be easily obtained from a suitable
arrangement of the first eight digits of our numeral system and their multiples, provided one
knows the right units to use.
Consider Table 3a. Each row after the first one is formed by multiplying the first row times the
row number. This way, the second row is twice the first one, the third one is three times the first
one, and so on. The next step is to reduce each number to a single digit by consecutive
summation, sometimes also called distillation (Table 3b). We have highlighted the key cells in
Table 3b that will allow us to calculate the dimensions of the Great Pyramid.
http://www.gizapyramid.com/Great%20Pyramid%20Architect%20Had%20A%20Secret.pdfhttp://www.gizapyramid.com/Great%20Pyramid%20Architect%20Had%20A%20Secret.pdfhttp://www.gizapyramid.com/Great%20Pyramid%20Architect%20Had%20A%20Secret.pdfhttp://www.gizapyramid.com/Great%20Pyramid%20Architect%20Had%20A%20Secret.pdf8/12/2019 Phi in the Great Pyramid
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(a) The first eight numbers and their multiples. Eachrow is the first one times the row number.
(b) Reduction of the numbers in preceding table to asingle digit. We have highlighted the numbers that allowto calculate the dimensions of the Great Pyramid.
Table 3
Now the cubes of the four numbers in any of the surrounding lines (Table 4a) are added to
obtain the first key number:
63+83+13+33=756
Do you recognize this number? Yes, using the feet as a unit, this is exactly the base length of the
Great Pyramid.
2b=756ft=440ec
(a) Adding the cubes of any of the highlighted lines (b) Adding the cubes of any line in the blue cross
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provides the base length of the Great Pyramid in feet. provides the exact ratio 4/r.
Table 4
Then any of the lines in the middle blue cross (Table 4b) provides the second key number, again
after summation of the cubes: 33+73+23+63=594
If this number does not have any meaning for you, just notice where the quotient of both leads
us: 756594=1411=4r
As you can see, this is exactly the ratio that allows to calculate the height of the Great Pyramid
given half its base: h=1411378=481.1ft=280ec
7. Conclusions
The Great Pyramid encodes much more information than the one summarized in this article. We
encourage the reader to make his/her own search for information. At a higher astronomical
scale, it is known that the Great Pyramid hides the grand cycle of Precession of the Equinoxes of
our solar system around the central sun of the Pleyades (25827.5 years) in many of its
dimensions (for example in the sum of the diagonals of its base expressed in pyramidal inches).
It is also well known that the three pyramids in the Giza complex are aligned with the stars in
the Belt of Orion. The interested reader is directed to the extense research of catalan
architectDr. Miquel Prezfor further information. It appears that we can draw a single
conclusion from all the preceding: the architects of the Great Pyramid of Giza were extremely
wise beings, with an advanced knowledge of math and astronomy far beyond the standard of
their time, and I would say even far beyond our own current knowledge.
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