The Myth:
The pyramids of Egypt, or more specifically, the Great Pyramid at Giza, was designed using the golden ratio.
The Reality:
One of the best discussion of this myth is in Markowsky's excellent paper [Mar92], though [Liv02] provides additional insight into the persistence of this myth. The source of this myth appears to be a highly imaginative translation of a description of the construction of the Great Pyramid in the History of Herodotus. This imaginative translation suggests that the dimensions of the pyramid were chosen so that the square of the height should be equal to the area of one of the triangular faces. Such a choice would dictate that the slant height of a face to half the length of the base would be the golden ratio. Unfortunately, this does not in fact appear to be what Herodotus said at all, and in fact the dimensions that Herodotus gives (roughly 800 feet to a side on the base and 800 feet tall) are wildly off.
My own view is that while it is true that the slant height of the Great Pyramid to half the length of the base does in fact come out to about 1.62 [Mar92] is a happy accident. Just a generation or so prior to the construction of the Great Pyramid we see the construction of the first true pyramids in Egypt (as opposed to the earlier step pyramids), and as the Bent Pyramid amply demonstrates, determining proportions and construction techniques for such mammoth stone structures was largely a process of trial and error, and the angle of the sides to the base was one of the variables being manipulated while they tried to figure out how to build them without suffering collapses, cracks and subsidence in the structure.
References:
[Liv02] Mario Livio, The golden ratio, Broadway Books, New York, 2002, The story of phi, the world’s most astonishing number. MR MR1938220 (2003k:11025)
[Mar92] George Markowsky, Misconceptions about the golden ratio, The College Mathematics Journal 23 (1992), no. 1, 2–19.
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