Comments on: The Parthenon and the Golden Ratio: Myth or Misinformation?

By: Gary B Meisner

Gary B Meisner — Sun, 10 Oct 2021 14:08:57 +0000

Great points, and I agree with all of them.

I wrote a full rebuttal in 2016 to Dr. Markowsky’s 1992 article here:

https://www.goldennumber.net/golden-ratio-misconceptions-by-george-markowsky-reviewed/

I don’t doubt that he had good intent, and give him credit for putting more thought into his article than the critics who just claim “No evidence exists!” Unfortunately, the article was biased, very poorly researched and had many flawed assumptions and conclusions.

More unfortunately yet, so many people, most notably academics, have quoted the article without applying a bit of critical thinking or analysis to it. Isn’t peer review supposed to be the hallmark of academic research that assures the quality of academic literature?

By: R. Mashlan

R. Mashlan — Sat, 09 Oct 2021 23:02:32 +0000

Have you thought about writing a full rebuttal to Dr. Markowsky’s 1992 article “Misconceptions about the Gold Ratio”, which seems to be the often cited source of the cynicism about the use of the Golden Ratio in ancient architecture?
I feel like his attack on the Kepler Golden Triangle in the design of the the Great Pyramid was very dismissive of the ancient Egyptian’s abilities as mathematicians. Unfortunately, we do not have the blueprints that they used (as papyrus tends to disintegrate after a millenia or two) but a millennia after the pyramid was built, we know of the Rhind Papyrus, which entailed mathematics useful for economics and tax collectors, including a fairly sophisticated approximation of pi that differed by only one percent, The Sumerians, only one river basin away from Egypt, and of the time period that is roughly contemporary with Khufu (circa 2600 BCE), we knew had methods of dealing with irrational numbers and calculating approximations for a positive root of quadratic equations, from a tablets dated to around 2000 BCE. (6 centuries after Khufu, 16 centuries before Pythagoras discovered irrational numbers, and over 20 centuries before Abu Kamil systematically solved quadratic equations using al-jabr)
We have evidence that the ancient Egyptians of Khufu’s time were excellent astronomers, which requires a deep understanding of mathematics. I would think that in building the world’s tallest building (that held on to that record for almost four millennia until the 14th century) would have been very special, and they would have put their best and brightest math nerds on it.

By: R. Mashlan

R. Mashlan — Sat, 09 Oct 2021 05:27:29 +0000

Pythagoras, about 50 years before construction of the Parthenon, invented the framework of ancient Greek geometry, when the Pythagoreans discovered the existence of irrational numbers and wanted to study them. The symbol of the Pythagoreans was the regular pentagram, a geometric figure that embodies the Golden Ratio like no other. His teachings were the likely source for Euclid’s Elements books I and II, Book II being titled “The Theory of Rectangles”., which has several instances of the Golden Ratio, starting with Proposition XI, where Euclid presents a construction of the Golden Rectangle.

There was a huge rift the ancient mathematics of Greece as this time, where the conservative side rejected the notion of irrational numbers and stuck to using rational numbers and arithmetic, and on the progressive side were those that embraced the new invention of geometry, which operated with continuous proportions and irrational numbers. This argument is reflected in the source material you quoted from Flon.

I would suspect that the progressive geometers won out over the conservative arithmeticians on the design of the Parthenon, intended to be the grandest building building in Athens.

By: Gary B Meisner

Gary B Meisner — Mon, 26 Apr 2021 01:16:42 +0000

In reply to Eleftherios Pavlides.

We shouldn’t be so quick to assume that Phidias had nothing to do with the golden ratio. Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (flourished c. 440 BCE). Phidias lived from 480 BC to 430 BC. Wikipedia reports that “is possible that most of sculptural decoration of the Parthenon was the work of Phidias’ workshop.” (https://en.wikipedia.org/wiki/Phidias)

It’s true that the Parthenon has “no straight lines,” but the curves are almost visually imperceptible and used only to improve the visual appearance of the structure. See my analysis at https://www.goldennumber.net/parthenon-golden-ratio-design/ and you’ll find plenty of examples of the golden ratio in its design. Furthermore, it’s actually quite easy to apply the golden ratio to a curved line. The “golden angle” that is the basis for the spirals that appear in plants is determined by taking a 360 degree circle and dividing it by the golden ratio to get two arcs of 137.5 degrees and 225.50 degrees.

Re “real artists do not paint by the number,” Kenneth Clark, author of “Civilisation” observed “This union of art and mathematics is far from our own way of thinking, but it was fundamental to the Renaissance.”

By: Eleftherios Pavlides

Eleftherios Pavlides — Sun, 25 Apr 2021 13:02:38 +0000

Of course Phidias had nothing do with the golden section that was described by Euclid a generation after the construction of the Parthenon. It was not named “golden” until some procrustean German named it “golden” in the 19th century. Phidias did not like straight lines and the Parthenon had no straight lines. How can you have a golden section defined on a curved line?

I called procrustean the naming of the a/b=b/(a+b) ratio golden, because it is trying to shoehorn beauty into math. Real artists do not paint by the number. It is the other way around they discover numbers through their art.

By: Marshall Walker

Marshall Walker — Tue, 26 Jan 2021 16:54:01 +0000

I have been thinking of the about patterns in nature and the golden ratio for a long time. Your article addressing the controversy around the parthenon is appreciated. I am currently writing a book on the foundations of pattern in nature and I have tentatively abstracted some images of the parthenon from from your article. which I attribute to you. If and when this come to fruition, I of course would seek some formal approval

By: Gary B Meisner

Gary B Meisner — Wed, 16 Sep 2020 20:56:35 +0000

In reply to David Youse.

But no one would claim that a building with “some proportions” between 1.5 and 1.7 has therefore “conformed” to the golden ratio. We’re talking about a building which has nine of its most fundamental design features reflecting the golden ratio to at least 3, if not 4, places of accuracy.

To keep things simple and conservative, let’s just say six key elements of the Parthenon’s design reflect the golden ratio, accurate to 3 decimal places. There are 21 three-digit numbers from 1.5 to 1.7 (e.g., 1.50, 1.51, 1.52, etc.) If all we were trying to do is find proportions on a building between 1.50 and 1.70, we’d have a 1 in 21 chance of any proportion hitting 1.62 (the rounded golden ratio). For six primary design elements to come in at 1.62, however, the odds of that are 1 in 21 to the sixth power, or 1 in 85,766,121.

What basis in reason or mathematics is there for concluding that this “occurs by chance?” Doesn’t the evidence more reasonably lead to the conclusion that the architects were intentionally incorporating the golden ratio into the design, and not just randomly picking “visually pleasing” proportions between 1.5 and 1.7 as they went along?

By: David Youse

David Youse — Wed, 16 Sep 2020 07:17:04 +0000

I think that one could look at just about any building on earth and find some aspects of it the conform (almost) to the golden ratio. This does not in any way mean that the ratio was used in its design, it simply means that some elements are in a ratio between 1.5: to 1 and 1.7 to 1. It’s simply a visually pleasing ratio that occurs by chance.