Posted by: Dave Richeson | April 27, 2011

## A pyramidologist’s value for pi

Recently I came across two theories about the design of Great Pyramid of Giza.

• If we construct a circle with the altitude of the pyramid as its radius, then the circumference of the circle is equal to the perimeter of the base of the pyramid. Said another way, if we build a hemisphere with the same height as the pyramid, then the equator has the same length as the perimeter of the pyramid.
• Each face of the pyramid has the same area as the square of the altitude of the pyramid.

Apparently these are favorite mathematical facts (especially the first one) for pyramidologists who look for mathematical relations in the measurement of the pyramids that help justify their cultish belief in the mystical power of the pyramids.

Of course we should separate the mathematical properties of the pyramids that may have been legitimate design decisions by the architects, from the crazy meanings that are often attached to them. I have no training in the history of Egyptian mathematics or in the history of the pyramids, so I can’t really assess their likelihood of being true (my guess: the first one is an amazing coincidence, the second is more likely to be intentional). However, one interesting fact is that if the first one was intentional, then they were using the value 3.143 for pi, which is significantly better than the value found in the Egyptian Rhind papyrus (3.16), which was written 600-800 years after the construction of the pyramids.

Just for fun, here are a few mathematical exercises:

1. Check these facts using the actual measurements of the pyramid (you can take altitude to be 146.6 meters and the length of one side of the pyramid to be 230.4 meters). They are indeed remarkably close!

2. Assume that the first one is true. Use the measurements given in (1) to show that the architects were using the value 3.143 for pi.

3. Assume that we have a pyramid for which both of these facts are true. Show that this would imply that

$\pi=2\sqrt{2\sqrt{5}-2}=3.1446\ldots$

Has anyone seen this approximation for pi before? I didn’t find it after performing a quick search of the internet. [Update, another way of writing this approximation is $4\sqrt{1/\varphi}$, where $\varphi$ is the golden ratio.]

[The photograph of the Pyramid of Giza is from Wikipedia.]

## Responses

1. Interesting post! Also, I enjoy reading this blog a lot!
About the pi approximation at the end, I did a little research on it and here is what I’ve found:
The last equation can be written as pi = 4*sqrt(phi-1) , which can also be written as 4/pi = sqrt(phi)
According to Wikipedia, this relation is coincidental:
http://en.wikipedia.org/wiki/Golden_ratio#Mathematical_pyramids_and_triangles

2. show that the architects were using the value 3.143 for pi.

is it more likely that they were “using the value 3.143″, or that they
were using “2 wheel diameters tall, and one wheel rotation across” and
the deviation is due to imprecise construction techniques?

3. From my calculations I believe this is indeed the true value for pi.
Panagiotis Stefanides demonstrates two examples of this this quite clearly at his website:

http://www.stefanides.gr/

• Dear Arunski I thank you.

Panagiotis Stefanides.