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© Copyright 2001, Jim Loy
In Pi and the Great Pyramid we see that the builders may have intentionally incorporated pi into the Great Pyramid (the width/height=pi/2, approximately), or maybe not. If it is a coincidence, just how strange a coincidence is it? I have decided to perform an experiment. Below, we see fifteen pyramids, with different ratios of height to width (different slopes). And we see some coincidences concerning many of those pyramids and certain famous numeric constants (roughly within 0.01). Phi is the golden ratio, and sqr() is the square root function.
| height/width | coincidences |
| 0.300 | slanting height/width=Euler's constant |
| 0.325 | height/width=1/pi |
| 0.350 | slanting height/width=1/phi |
| 0.375 | slanting height/width=1/phi |
| 0.400 | height/width=phi/4; diagonal of base/height=pi/11 |
| 0.425 | height/diagonal of base=1/sqr(11) [very accurate] |
| 0.450 | diagonal of base/height=pi |
| 0.475 | diagonal of base/height=pi [very accurate] |
| 0.500 | slanting height/height=sqr(2) [exact] |
| 0.525 | height/width=pi/6 |
| 0.550 | slanting height/height=e/2 |
| 0.575 | height/diagonal of base=phi/4 |
| 0.600 | height/diagonal of base=phi/4; slanting height/base=pi/4 |
| 0.625 | height/width=1/phi |
| Great Pyramid (0.636) | width/height=pi/2; slanting height/width=phi/2 |
| 0.650 | slanting height/height=sqr(phi) |
| 0.675 | slanting edge/width=pi/2 |
| 0.700 | slanting edge/width=pi/2 |
| 0.725 | slanting height/width=8/9 |
| 0.750 | slanting height/width=e/3 |
| 0.775 | slanting edge/width=phi |
| 0.800 | slanting edge/width=phi; slanting height/height=sqr(pi) |
| 0.825 | height/diagonal of base=Euler's constant |
| 0.850 | slanting edge/width=sqr(e) |
| 0.875 | slanting edge/height=phi |
| 0.900 | diagonal of base/height=pi/2 |
| 0.925 | slanting height/width=pi/3 |
| 0.950 | width/height=pi/3 |
| 0.975 | slanting height/height=sqr(phi) |
Here is a drawing of the first (0.300) and last (0.975)
pyramids above. From the table, we see that we can find coincidences for
virtually any pyramid. All of these coincidences are fairly accurate (within
0.01), especially if my pyramids are made of rough rock and the original
dimensions can only be guessed at. I've added the Great Pyramid (between 0.625
and 0.650), for comparison.
Above, I have only examined ratios of lengths, as they are unitless numbers. I have ignored coincidences involving area and volume, as such coincidences would normally depend upon the unit used. I have also ignored comparisons with very great lengths like the size of the earth, or the distance to the sun, as my pyramids are of indeterminate size.
The famous numbers found in my pyramids are (approximately): pi=3.14159, phi=1.61803, e=2.71828, Euler's constant=0.55722 (also called the Euler-Mascheroni constant), and the square root of 2 is 1.41421.