Your comments are spot on:

– Euclid’s Elements was THE geometry textbook for centuries after it was written in about 300 BC, only being replaced by other geometry textbooks in modern times.
– Elements has at least 10 propositions that describe the cut of extreme and mean ratio, aka the golden ratio.
– The divider has been a common tool of artists and artisans since antiquity for measurements and proportions.
– All it takes to create a golden ratio caliper is to join the two arms of a divider at the golden ratio cut described in Euclid’s Elements. No “construction” with compass and straight edge is needed, except once for the purpose of constructing the golden caliper.

It does seem inconceivable that best artisans working on the most important projects of their culture would not have realized and used this, especially when the evidence exists in so many works that it was known and applied with intent and accuracy.

If you’re not familiar with this man’s work, I think you’ll find it interesting or at least remarkable. Kind of represents the absolute opposite of what Markowsky tried to do by acidifying the omnipresence of a sublime principle by his sour soul.

https://www.youtube.com/watch?v=6DleAx-bxIs&t=3s

and a more in depth presentation:
https://www.youtube.com/watch?v=upzjM7-83LE

Thanks, a true joy to read

Thanks for your comments, Remi. A few responses:

The disturbing thing to me about the Herodotus quote is that the validity of the quote was used to dismiss statements about the pyramids rather than the actual measurements of the pyramids.

It will could be that the 5.5 seked was used, but the question is then WHY that ratio rather than some other, especially when it is about as mathematically as close as you can get using integer numbers to the actual golden ratio.

The Parthenon is known to use many numbers and ratios, not just 16/9, and my measurements of high resolution images in my article at https://www.goldennumber.net/parthenon-phi-golden-ratio/ show proportions of design that are very closely aligned to Phi (1.618…), and nowhere near 16/9 (1.777…)

Measurements on artwork cannot “prove” that the golden ratio was applied by the artist, at least not without a written testimony by the artist, but repeated applications of the the golden ratio within a composition with as much accuracy as can be achieved is certainly very strong and compelling evidence. And given the evidence, one has to ask why it is logical or reasonable to assume that the artist did not apply the golden ratio. It’s been known about since the time of the ancients, and was covered explicitly by Euclid in a number of his constructions in Elements, a book that one of the most translated and distributed in all of history. It was THE geometry book from which all students learned until other texts on the subject began to be written only in the past couple of centuries.

Please refer to relevant work:

http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf

http://www.stefanides.gr/pdf/2012_Oct/PHOTO_13_GEOMETRY_DESIGN.pdf

Regards from Athens,

Panagiotis Stefanides

Thank you Juan for the supportive words. You’re correct that the ratio of hypotenuse of 612.005 to the height of 481.4 is 1.2713. That is not the measure in question though. A perfect phi-based triangle has these proportions:

Base – 1
Height – Square root of Phi = 1.27201965…
Hypotenuse – Phi = 1.61803399…

So the ratio of the hypotenuse to the height AND the height to the base are both 1.27201965, the square root of two. In the Giza pyramid the ratio of the hypotenuse to the height is 1.27130 and the height to the base is 1.27390.

The variance of the Giza pyramid to the Phi pyramid is 1.27130/1.27201965 or 1.27390/1.27201965, a variance of 0.1% in both ratios. The hypotenuse of the Giza pyramid is 1.61951173 times the base, while the hypotenuse of the Phi pyramid is 1.6180339887 of it’s base. This is a variance of only 0.09%.

Hello, This article is awesome and I applaud and praise you for it, but not trying to be malicious or negative about this comment I think you have a mistake in your calculations of the ratio of the hypotenuse to the height in your pyramid portion of your article. If the height is 481.4 and the Hypotenuse is 612.005 then shouldn’t the answer be 1.2713? I have checked it with several calculators and my laptop. Sorry.

JMM