The Great Pyramid, The Great Discovery, and The Great Coincidence
Posted by Darshana Sanrakshak Shambhala on January 2, 2011 at 9:57pm in The Egyptians
by
Mark Herkommer
June 24, 2008
The Great Pyramid
The
Great Pyramid, The Pyramid of Cheops (Khufu), built between 2589 - 2566 BC,
stands today as one of the greatest architectual achievements. Its dimensions
and orientation have astounded mathematicians and astronomers for centuries.
Many subscribe to the belief that only God or god-like space aliens could
constructed it. I hope to persuade you, gentle reader, to the
contrary.
In my opinion it is time to take a good hard look at the
Egyptians and their sciences and try to draw some rational conclusions. I belive
that the Pyramid was constructed by the Egyptians and their army of laborers.
This was done without any extraterrestrial assistance or metaphysical
manifestations. Clear thinking and hard work, a little good fortune, and an
interesting mathematical coincidence, but not much more.
The skill of
their masonry is beyond dispute. The pyramid was built in about 23 years with
about 2.5 million stones. This means that every year 100,000 large blocks (285
per day), weighting an average of 2 1/2 tons must have been quarried, dressed,
brought to the building site, and set in place.
Like so many of the
baffling ancient monuments, no nation today on Earth has the technology required
or the capital wealth to attempt to duplicate the majestic works of these lost
civilizations.
Things That Ancient Egyptians Knew
From all
accounts, the Egyptians were meticulous record keepers. From dynastic history to
harvest yield to small transactions, ample evidence exists to demonstrate their
ability and inclination to keep records.
One can speculate as to the
basis for this cultural characteristic: Nile flooding, pharaohic rule, general
commerce, etc. One this is certain; contemporaneous cultures did not keep
records to the extent that the Egyptians did.
Every civilization has its
share of creative individuals. These individuals borrow from their culture and
synthesize new information. From the detailed historical records, we know that
over time the ancient Egyptians developed a body of abstract information.
Historical investigations into the records and buildings of the Egyptians has
established they developed and maintained a body of scientific information.
These things the ancient Egyptians knew with certainty (in no particular
order):
* how to manipulate rational numbers (to some
extent)
* how to bisect an angle
*
how to construct similar polygons of certain ratios
* how
to find vertical and horizontal
* that a triangle with the
sides 3:4:5 was a right triangle
* that the length of a
year to be 365 days
Additionally they probably
knew:
* the cardinal points of the compass as we
understand them
* that the earth was a sphere of immense
size
* that certain other triangles having integral sides
were also right triangles ( just as the ancient Babylonians
did)
* that there existed numbers that could only be
approximated, such as Ö2, Ö3 and f (the Golden Ratio).
What these simple
lists suggest is that the ancient Egyptians understood relationships between
abstract ideas, even though the computational machinery had not yet been
developed. As algorithms were invented the Egyptians used them to produce
"mathematical handbooks". One example of this is the Rhind Papyrus.
The
Great Discovery: 2pr = c
At some point in their history, probably during
the 4th dynasty, the ancient Egyptians came to a stunning
realization:
the ratio of a circle's circumference to its diameter is a
constant
Once recognizing this fact, I am sure it would not be long
before they would try to estimate p, probably as 3 1/7. I also believe that they
could easily discover that this is only an approximation; that the "true" value
of p is more elusive.
On the other hand, just because they did not have a
more exact value for p did not mean that they could not fully exploit it. Their
love of the circle is evident in much of their worship, art, glyphs, and
architecture. The Egyptians were truly a circumphillic society.
p in the
Great Pyramid
In honor of their great discovery, the circumphillic
Egyptians found a way to incorporate p into the architecture of Great Pyramid
through a simple yet elegant vehicle:
2pheight = perimeter
This of
course is same formula expressing the relationship between a circle's diameter
and its circumference. What is even more interesting is that they did not even
need to calculate p to employ it.
By measuring using a wheel having the
diameter of 2 units (cubits), they could roll out the perimeter exactly as many
revolutions as the pyramid was units tall. Although p was never specifically
calculated, its value became incorporated to a very high degree of accuracy. By
some estimates, five decimal places. WOW!!!
Pyramids Are Monuments to
Mathematical Knowledge
How can I assert that the Great Pyramid is a
monument to p? If you look at the other finished true pyramids (not step
pyramids) of Egypt, you can see the Great Pyramid is one of three that embodies
p. These three pyramids, which I shall call "p pyramids", were apparently
constructed around the 4th dynasty (ca. 2575 to 2450 BC).
Other
mathematical relationships appear to be represented as well. Any relationship
that is not within one minute of arc (1/60º) is marked with "?":
pyramid
location dynasty
base
(meters) height
(meters) face
angle
(º) mathematical
relationship
Snofru
Maidum 3rd/4th 147
93.5 51.8431 p
pyramid
(renovated step pyramid)
Snofru - "Bent Pyramid"
Initially
built as a equilateral pyramid,
the lower section was girdled and the
upper
section completed with a lower slope angle. Dahshur
4th 188 105
initial: 60.0000
lower: 54.4622
upper: 43.3667
initial: equilateral
lower: (unknown)
upper: 20:21:29
pyramid (?)
Snofru - "Red Pyramid" Dahshur
4th 220 104
43.3667 20:21:29 pyramid (?)
Menkaure'
Giza 4th 105
65.5 51.3403
(unknown)
Khufu (Great Pyramid) Giza
4th 230 146 51.8431
p pyramid
Khephren Giza
4th 214.5 143.5
53.1300 3:4:5 pyramid
Sahure'
Abusir 5th 78.5
47 50.1944
(unknown)
Neuserre' Abusir 5th
81 51.5 51.8431
p pyramid
Neferirkare' Abusir
5th 105 70
53.1300 3:4:5 pyramid
Userkaf
Saqqara 5th 73.5
49 53.1300 3:4:5
pyramid
Wenis Saqqara 5th
57.5 43 56.3097
20:21:29 pyramid on diagonal (?)
Izezi
Saqqara 5th 78.5 52.5
53.1300 3:4:5 pyramid
Teti
Saqqara 6th 78.5
52.5 53.1300 3:4:5
pyramid
Pepy I Saqqara 6th
78.5 52.5 53.1300
3:4:5 pyramid
Merenre' Saqqara
6th 78.5 52.5
53.1300 3:4:5 pyramid
Pepy II
Saqqara 6th 78.5
52.5 53.1300 3:4:5
pyramid
Senwosret III Dahshur 12th
105 78.5 56.3097
(unknown)
Amenemhet III Dahshur
12th 105 81.5
57.2639 65:72:97 pyramid (?)
Amenemhet
I el-Lisht 12th 78.5
55 54.4622
(unknown)
Senwosret I el-Lisht 12th
105 61 49.3986
(unknown)
Amenemhet III Hawara
12th 100 58
48.7500 Golden Rectangle on diagonal
(?)
Senwosret II el-Lahun 12th
106 48 42.5833
(unknown)
Khendjer Saqqara
13th 52.5 37
55.0000 (unknown)
There are a
number of queen's pyramids that have the same dimensions and face angle, showing
a consistency in aesthetic design and building practices.
It is important
to observe that there a general climb in face angle over each of the dynasties,
but it is not incremental. Early attempts in the 4th dynasty to increase the
face angle to 60º failed on at least two occassions.
Most striking is the
repeated face angle of 53.1300º, which is the angle created by a triangle having
the ratio 3:4:5. This face angle was typical of pyramids constructed during the
6th dynasty (ca. 2350 - 2180 BC).
Similarly, the incorporation of p into
the Great Pyramid (Khufu) is almost certainly a deliberate act. Why would
Snofru's pyramid in Maidum, first built as a step pyramid in the 3rd dynasty, be
renovated to incorporate p? Further, only one other p pyramid was built after
the Great Pyramid, Neuserre's at Abusir.
The fact that there are only
three p pyramids may may speak to the computational difficulties faced in
attempting a construction an irrational number as the base:height ratio for the
pyramid. It would be clear to the Egyptians that the 3:4:5 ratio is much easier
to scale to the face construction and follows the trend of climbing face
angles.
The Snofru pyramid at Dahshur (4th Dynasty) has a face angle of
43.3667, which approximates within ¼º the angle of a 20:21:29 triangle
(43.6028º). Other pyramids are within a similar margin of error. Because this is
a very early true pyramid, it may be that the Egyptians had not yet mastered the
process of architecting certain known mathematical relationships into their
pyramids. Alternatively it may be based on some other yet unrecognized
mathematical fact.
Two other interesting pyramids from the 4th Dynasty
are the the "Bent" Pyramid at Dahshur and the Ra'djedef pyramid (The "Lost"
Pyramid) at Abu Rawash. The Bent Pyramid was originally constructed with a face
angle of 60º. Due to apparent structural instability was girdled with a casing
having a 54.46º face angle and finished finished with the upper section having a
43.37º face angle. The Ra'djedef pyramid, although unfinished (or possibly
destroyed), has an apparent face angle of 60º. In medial cross-section these
pyramid would have been an equilateral triangle; I call this type an equilateral
pyramid.
Both pyramids reflect another mathematically important triangle,
the equilateral triangle. The equilateral triangle, with its three equal length
sides, is easily constructed, architectually versatile, and is aesthetically
pleasing. Although probably not part of the ancient Egyptian mathematical
heritage, can be thought of as two right triangles that have the proportions
1:Ö3:2, usually referred to as a 30:60:90 triangle.
These facts are more
than mere coincidences. It is clear that the Egyptians incorporated their
acquired fundamental mathematical knowledge into the design of their pyramids.
Further, the incorporated mathematical relationships have share an important
characteristic with the Egyptian Pharaoh perception, both are eternal. What
better way to recognize the spiritual immortality of a Pharaoh than with
immutable enduring mathematical truth?
The Great Coincidence
What
the ancient Egyptians almost certainly did not know is a particular mathematical
coincidence involving p and f, the Golden Ratio. It is this coincidence that has
in large part fueled speculation about outlandish origins of the Great
Pyramid.
Recall first that the Golden Ratio is the solution to the
quadratic: f + 1 = f2. The solution is the irrational number: f = (1 + Ö5)/
2.
Its is unlikely that the ancient Egyptians could solve this since they
did not have mathematical procedures to solve quadratics. However, the equation
itself may have been known to them from its geometric interpretation in the
Golden Section. It is worth noting that the Golden Section did not make a
possible appearance in pyramid until the Amenemhet III pyramid at Hawara of the
12th dynasty (ca. 2000 - 1800 BC).
The Great Coincidence is this:
p /
2 » 2 / Öf
Compare the values from either side of the equation:
p / 2
= 1.570796...
and
2 / Öf = 1.572302...
A difference of merely
0.001506... . (Note that since the latter expression involves the square root of
the Golden Ratio, which itself contains a square root; the final quantity
therefore involves finding a quartic root.)
Why I call this the Great
Coincidence is because now it appears as though f is also incorporated into the
Great Pyramid as well.
With a little algebraic manipulation we can see
that the apothem is in proportion to the base by the Golden Ratio. If we take
the base/2 = 1, then we have a triangle with apparent proportions base : height
: apothem as 1 : Öf : f.
But was this part of the design? No, it was
not.
Another interesting relationship occurs because of this coincidence.
The area of the face of the pyramid is equal to the height squared.
As
before, let base/2 = 1, then using the formula for the area of a
triangle:
1/2 * base * apothem = f
Of course this is equal to the
height squared:
height * height = Öf * Öf = f
But was this part of the
design? Again no, it was not.
All we really have in the ingenious
incorporation of one universal constant in a monumental architectual work, and
an interesting mathematical coincidence. Furthermore, any p pyramid of any scale
must necessarily incorporate these ancillary geometric relationships.
It
may come as a disappointment to some that extraterrestrials (or God) did not
need tell the Egyptians the value of p or f. The truth is that extraterrestrials
(or God) have no more fingerprints on the construction of the Great Pyramid than
you or me.
Disappointed? Don't be. Humanity should take pride in the
manifest intelligence of ancient civilizations and purchase hope for future
endeavors. I know I do.
One Final Note About The Amazing Alignment Of The
Great Pyramid's Base
Many people find it remarkable that the Great Pyramid's
base is aligned almost perfectly with true North. In point of fact, for a
surveyor this is not a great feat at all. True North can be found with a stick
and a circle and the knowledge of how to bisect an angle.
The following
procedure will very accurately locate true North. The longer the stick, the
greater the accuracy:
1. Plant the stick vertically in the
ground.
2. Draw a circle with the string roughly the same radius
as the stick is long.
3. Watch the path of the sun during the
day. Mark where the sun's shadow cast by the very tip of the stick crosses the
circle.
4. Make an angle using the sun path intersections and
the circle's center
5. Bisect the angle ~ the bisector points to
true North.