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The ratio of four over the square root of the Golden Mean is very close to $\pi$.

Therefore, if the height of a pyramid of square basis of size 2L is L times the square root of the Golden Mean, the perimeter 8L of the basis is almost equal to the perimeter of the circle the radius of which is the height.

But it also means (using Pythagoras and the definition of the Golden Mean) that the triangle that forms the face of the pyramid has height L times the Golden Mean, which is, aesthetically, a desirable ratio.

Now it is a fact that this is very close to the proportions of the Great Pyramid of Gizeh.

In his book Les nombres et leurs mystères, André Warusfel claims that the Egyptians did not care about $\pi$, their only interest was in the Golden ratio. But is that true?

Are there any archeological arguments to decide between the four options :

  1. Warusfel's one : The Egyptians cared only about the Golden Mean, not $\pi$

  2. The Egyptians wanted the perimeter of the pyramid to coincide with the perimeter of the circle with its height for radius and did not care about the proportions of the faces

  3. They wanted both things, because they were convinced that $\pi$ was exactly $4 \sqrt{\frac {\sqrt 5-1} 2 }$

  4. They did not care about either of these and just took for the ratio of the height to the side of the basis the simple ratio 7/11, which is also very close

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  • $\begingroup$ I’m voting to close this question because it is not a question about the history of science. Of Egypt, maybe, or of the pyramids, but not of science. $\endgroup$ Jan 9 at 0:31
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    $\begingroup$ @kimchilover Well, it is a historical question that has to do with mathematics. Why would you say it is off-topic ? And there could be archeological evidence for other mathematical knowledge of the Egyptians that could shed light on it. $\endgroup$
    – Alfred
    Jan 9 at 1:39
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    $\begingroup$ This is mostly reading faces in the clouds and tea leaves. One can conjure up all sorts of "almost relationships" involving any sorts of numbers in multiple ways, and then "find" them in multiple places. Seewald's website documents this sort of activity with the golden ration in Giza and other places. $\endgroup$
    – Conifold
    Jan 9 at 4:53
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    $\begingroup$ See Misconceptions about the golden ratio by George Markowsky (1992). $\endgroup$ Jan 9 at 7:17
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    $\begingroup$ Most of the smooth facing stones have been removed from the pyramids at Giza, leaving rough hewn stones behind. The upper few dozen feet of the Great Pyramid are mission, perhaps because they were covered in gold leaf. So to me it seems very hard to make precise measurements of the proportions of the Giza pyramids. Note that the Bent Pyramid has a steep lower part and a flatter upper part, showing that Egyptians didn't always use the same angle on pyramids. $\endgroup$ Jan 9 at 17:47