## My Pyramids

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.