(Some Elegant Numerical Relationships)

Some time ago I devoted a few hours to playing with the dimensions of the
Great
Pyramid at Giza as an experiment with numbers. Some of the relationships that are mentioned below are quite famous, such as the ones involving the numbers ** &pi** and

Some scholars have investigated the question of what the intentions of the
architects involved in building the various pyramids might have been. For example, several pages in an article ** Mathematical Bases of Ancient Egyptian Architecture and Graphic Art** by G. Robins and C. Shute (

According to __Mysteries of the Great Pyramids__ by A. Pochan, the angle that each face of
the Pyramid makes with the base is approximately 51.85^{o}.
To make the numbers precise in the following experiment, I decided to consider a mathematically perfect
pyramid with a square base, isosceles triangles as faces, and with
51.85^{o } as the exact angle which each face makes with the
base. I then searched for numerical relationships involving
the following six lengths incorporated in this pyramid.

s = the length of one side of the base

p = the perimeter of the base

d = the length of a diagonal of the base

h = the height of the pyramid

r = the length of one edge (or ridge) of the pyramid

f = the length of the line joining the midpoint
of

one side of the base to
the apex.

To make this clear, each of these quantities represents a distance. If you stand at one corner of the base of the pyramid, s is the distance from you to the next corner, d is the distance from you to the opposite corner, and r is the distance from you to the apex of the pyramid. If you stand at the base of the pyramid halfway between two corners, then f is the distance from you to the apex of the pyramid. If you stand inside the pyramid at the very center, then h is the distance from you to the apex, which is of course just the total height of the pyramid. The perimeter p of the pyramid is the total distance completely around the base: p = s+s+s+s = 4s. The diagonal d is related to the side s by d = \/2 s . If we take the length s of one side of the base as the unit of measurement, then by using some trigonometry we obtain the following values (to six decimal places):

s = 1 , p = 4 ,
d = 1.414213..., h = .636528...,
r = .951403..., f = .809425....

Below are some of the relationships which are exhibited
by the pyramid. Relationships 1, 2, 3 and 5 are discussed in Pochan's book mentioned above. They are also discussed in Peter
Tompkin's book __Secrets of the Great Pyramid__, especially the very famous relationships involving the numbers ** &pi** and

1.
p /
h
~
2*&pi**
*( .015% )

2. p / h ~ 2(22/7) (.026%)Herepi= 3.141592.... For more about the number, visit the The&piPiPages. This relationship means that the perimeter of the Great Pyramid is approximately equal to the circumference of a circle of radius h.

This relationship is related to the first one because 22/7 is a famous classical approximation to. In fact, the ratio p/h is somewhat larger than 2&piand somewhat smaller than 2(22/7). An equivalent way to state this relationship is in terms of the slope of each face of the Pyramid: h /(.5s)=14/11. If you start from the center of one side and climb directly toward the apex of the Pyramid, then for every 14 feet higher (vertically) above the base you get, you will move approximately 11 feet closer (horizontally) to the center.&pi

3.
h /(.5d)
~ 9/10
( .02% )

The ratio h/(.5d) is the slope of each edge of the Pyramid. This relationship means that if you are climbing up one of the edges of the Pyramid, then for every 9 feet higher above the base that you climb, you will move approximately 10 feet closer to the center.

The inverse of this slope is 10/9=1+1/9.

4.
f /(.5s)
~
F_{9
}/F_{8 }
( .015%)

5. f /(.5s) ~Here F_{8}= 21 and F_{9}= 34 are the 8th and 9th Fibonacci numbers.

Here= (1+ \/5 )/2 = 1.618033..... is the famous Golden Mean. One should compare this relationship to the more accurate relationship #4. It can be proved that&phican be approximated extremely accurately by the ratios of consecutive Fibonacci numbers. In fact, in a certain precise sense, the best approximations to the irrational number&phiby rational numbers are given by the rational numbers of the form F&phi_{n+1}/F_{n}, where F_{n }is the n-th Fibonacci number and F_{n+1}is the (n+1)-st Fibonacci number. It just so happens that f/(.5s) is fairly close to, but even closer to the nearby rational number F&phi_{9}/F_{8}.

6.
r / f
~ e^{2 }/
2* &pi
*(.06%)

Here e = 2.718281.... is an extremely important constant which was introduced into mathematics in the 18th century. For various ways to define e, visit here.

7.
s /
r ~
1^{ }+ 1/3^{3}
+ 1/5* ^{3}*+
1/7

8. 3s / h ~ e /The infinite series which is on the right-hand side of this relationship converges to a certain number, but mathematicians have been able to prove very little about this number. It probably cannot be expressed in any simple way in terms of, e, or other well-known numbers. It is closely related to Apery's constant. In the late 1970's , a proof of the irrationality of this number was discovered by the French mathematician Roger Apery.&pi

Here= .577215.... is the "Euler-Mascheroni constant." The ratio 3s/h can also be written as s&gamma^{3}/^{1}/_{3}hs^{2}, which is the ratio of the volume of a cube built on the base of the pyramid to the volume of the pyramid itself.

9.
s /
h ~
e
/ \/3
( .1%)

10. h
/
f
~
* &pi*
/ 4
(.12%)

11. s
/
f ~ 1
+ 1/3^{2}
+ 1/5^{2}
+ 1/7^{2}
+ 1/9^{2}+
.....
(.14%)

COMMENTARY: We should emphasize that the relationships listed above are for the perfect pyramid where the base makes an angle of exactly 51.85This infinite series is known to converge to the number&pi^{2}/ 8 .

NorthWest | 89^{o}59'58" |

NorthEast | 90^{o}3'2" |

SouthEast | 89^{o}56'27" |

SouthWest | 90^{o}0'33" |

North Side | 230.253 meters |

South Side | 230.454 meters |

East Side | 230.391 meters |

West Side | 230.357 meters |

As these measurements show, the builders of the Great
Pyramid achieved a rather impressive level of accuracy, assuming that their
intention was to make the base a perfect square. The average length of
the four sides is 230.364 meters and the greatest discrepancy from that
average is .111 meters (the North Side), which is just 4 inches. The discrepancy
between the longest and shortest sides is just 8 inches. As a percentage
of the length, this is less than .09%. As for the angles, the SouthWest
angle is extremely close to 90^{o}, the percentage error being
just .01%. The largest discrepancy from 90^{o} is .065% (for
the angle at the SouthEast corner).

The
Great Pyramid is not a perfect pyramid in other ways too. The apex is missing.
The faces are slightly concave. As for the angle of 51.85^{o}
which we used in the above experiment, it is accurate to within an error
of about .03% according to Pochan. In Mark Lehner's authoritative
book __The Complete Pyramids__, the angle that each face makes with
the base is given as 51^{o}51'14" , which is just slightly more
than 51.85^{o} . In any case, the approximations that we
have given above will still be fairly accurate for the actual Pyramid,
but the errors may differ somewhat from those given above. Some of
the error estimates may turn out to be better, some worse.

But the most interesting question to ask is whether any of these relationships
were intentional. That is, did the Egyptian architect(s) have
one or more of these relationships in mind when designing the structure?
For some of the relationships given above, the answer is rather clear.
It is hard to believe that the Egyptians had any knowledge or awareness
whatsoever of numbers like e , ** &gamma**,
or the infinite
series occurring in #7 and #11. These numbers entered mathematics
during the 17th and 18th centuries. We have included them just to
make the point that it is not so hard to find coincidental numerical relationships
and their existence should not necessarily be regarded as significant.
But for some of the other relationships, it seems difficult to answer the
question with any confidence. Since there doesn't seem to be any
written documentation which directly explains the design of the Great Pyramid, one has
to attempt to be a mind-reader. I mentioned earlier the argument by Robins and Schute based on the Rhind Papyrus. I myself do not completely agree with their conclusion and would propose that both relationships #2 and #3 reflect the intentions of the architect. To be more precise, the architect may have chosen one of those relationships, cognizant of the fact that the other relationship would then be true. Both of these relationships may have been used in building the pyramid with accuracy. In the essay The Slopes of the Egyptian Pyramids, I will
present arguments in support of that possibility. As a consequence, it would seem reasonable to believe that all of the other relationships listed above, including the famous relationships
involving the numbers

COPYRIGHT © 2000 RALPH GREENBERG