Giza pyramids and convergents of √φ/2 — known connection?

Question

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The three main Giza pyramids have height/base ratios that are exact in royal cubits:

Pyramid	Height	Base	Ratio
Khufu (Great)	280	440	7/11
Khafre	274	411	2/3
Menkaure	125	200	5/8

These are the 4th, 5th, and 6th convergents of √φ/2 (where φ is the golden ratio):

Convergents of √φ/2: 1/2, 2/3, 5/8, 7/11, 159/250, ...

The √φ connection to the Great Pyramid's "Kepler triangle" geometry is documented (see R. Herz-Fischler, The Shape of the Great Pyramid, Waterloo, Ontario: Wilfrid Laurier University Press, 2000; or Nexus Network Journal 3, (2) 163, 2001).

Question: Has anyone noted that ALL THREE of the major Giza pyramids follow consecutive convergents of the same irrational number? Or is this observation new?

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Answer 1

You may want to consult two more recent books (both of which cite Herz-Fischler):

Mario Livio, The Golden Ratio, review, London, 2003.
Corinna Rossi, Architecture and Mathematics in Ancient Egypt, Cambridge University Press, first paperback edition, 2007.

Both discuss (and basically refute) the intended use of the Golden Section in the construction of the Egyptian pyramids as there is no (written) historical evidence. Livio points out that the use of $\pi$ is somewhat more likely as one can extract an approximate value from the Rhind papyrus (which is not true for the Golden Ratio, $\phi$). He also points out that length measurements (like those in your table) should be stated with error bars. For instance, the base lengths of the Great pyramids differ by about 0.67 feet which amounts to an error of roughly $10^{-3}$. Incidentally, this is the accuracy required to distinguish between Kepler's triangle (base/height = $2/\sqrt{\phi} \approx 1.572$) and a triangle with base/height = $\pi/2 \approx 1.571$ (which may result from using 'rolled cubits', see Livio). You have noted these very close values in your OP and comment.

Rossi's book is very useful as a repository for historical and other sources and as a data mine. In an appendix, she lists the dimensions and slopes of more or less all known pyramids. She also provides a classification in terms of the slope (angle) or the ancient Egyptian seked. A representative figure of Rossi's which includes your pyramids can be found below:

Years ago I reviewed Rossi's book for the Journal of Archaeological Science [36 (6), 1286 (2007)]. This is behind a paywall, but (if you are interested) a slightly updated versions can be found here.