Let me tell you the true story about the passage from Herodotus. I don't read a bit it of classical Greek, by the way. I got this from a source published before 1942. The material is very concisely presented but it is easy to identify the passage that the author is talking about. If you have an English version, very near where Herodotus talks about Cheops prostituting his daughter, there is a description of the dimensions of the pyramid (I gave the section no. in my #157). I have seen two distinct (supposedly anyway) translations - Great Books and Harvard Classics. Both of them say, essentially, that the height of the pyramid is equal to the length of the sides of the base, which is obviously false. The author of my source refers to an "obscure passage" (in the original Greek - but which must be the sentence that is translated as I describe above). He says that after a "minor literary emendation the passage makes perfect sense". And its substance is that the dimensions result in the relation that I quote: the area of a face equals the square of the height.
But there are a lot of unanswered questions: does the minor emendation actually restore the text to its original form that was corrupted by an error of some copyist? Was the relation above the original basis of the design of the pyramid, or is it simply (as you suggested it may be) an accidental relation that someone happened to notice in the 1000+ years between the pyramid being built and Herodotus' visit to Eygypt? I think the issue is interesting (and I sort of regret exploiting it (after a few beers) as a not very funny joke). I really wish I could find someone with a good knowledge of the classical Greek who would look at the passage and independently corroborate (or not) the analysis that I described above.
If you haven't worked out the math, the relation "the area of a face equals the square of the height" gives the following: the cotangent of the angle of elevation of a face is the square root of the so-called golden mean, i.e.
sq.rt.((sq.rt(5)-1)/2) = sq.rt.(.618034) =.78615
and the ratio of the perimeter of the base to the height(*) is 8 times that or
6.2892.
(*)It is of course that ratio which is approximately twice pi, not the inverse ration as I incorrectly wrote in my post #157,
At the time I read Herodotus (probably about 15 yrs. ago) I was not that interested in the GP so I paid little attention to the discussion of the dimensions.
However, I have a question for you that not even my Stat teacher girlfriend is able to answer. Did the Greeks have numbers? I have never seen a number in Greek. If we are familiar with Roman numbers, one would think we would have at least seen a Greek number. But I never have.