In 1959, Walt Disney produced a 27-minute educational film called “Donald in Mathmagic Land.” This memorable film was nominated for an Academy Award, and has been seen by countless students over the years. It explores the impact of mathematics in a variety of topics, including art, architecture, gaming, music, nature and the human body.
Donald discovers the Golden Ratio
At about 9 minutes into the film, Donald Duck visits the Parthenon of ancient Greece. Here, he is shown in a 15 second segment that its design embodies a number of golden rectangles. In the next 5 minutes, he learns about other appearances and applications of the golden ratio. Click below to watch on YouTube:
As popular as the film was in mathematics classes, it seems that some students who went on to earn their PhDs in Mathematics may have missed it, or perhaps they just disagree. On Wikipedia, for instance, they’re quoted saying things like “the entire story about the Greeks and golden ratio seems to be without foundation” and “such theories have been discredited by more recent studies, which have shown that the proportions of the Parthenon do not match the golden proportion.“
Could Walt Disney and Donald Duck have been so seriously mistaken? Did they lead millions of innocent school children astray? Let’s take a look at the evidence. Then you can decide for yourself whether Donald should be awarded an honorary PhD (a Ducktorate of Philosophy?) in Mathematics or be quickly discredited as a “quack.”
To do this, we’ll look at these golden rectangle overlays using high resolution photos of the Parthenon and the pixel-perfect measurements of PhiMatrix software to assess their accuracy.
A quick lesson in Doric architecture
Before we start, it may help to review the nomenclature of the elements of classic Doric architecture:
The Eastern Facade
Disney first shows that the height and width of the facade of the Parthenon form a golden rectangle. I presented the same finding in article my 2013 article on the Parthenon. I also found that the base of the entablature is at the golden ratio point of the Parthenon’s height.
It’s not clear though why the Parthenon’s architect would have used a golden rectangle that starts at the base of the second step. It might make a more compelling case to begin the golden rectangle at the base of the columns, or at the very first step. In any event, the Disney video shows a golden rectangle using same approach I used in my analysis. Let’s call this one intriguingly close, and see how the rest of the evidence stacks up.
Column to Capitals
The Disney video shows that another golden rectangle based on each pair of columns. The height of the columns in relation to the width of the capitals that crown each pair of columns form this golden ratio. I analyzed two high resolution photos of the Parthenon. On the Western facade, a golden rectangle aligns to the columns to the pixel. Curiously, the rectangle based on the center columns on the Eastern facade has a proportion closer to 1.645 than 1.618. This variation of about 1.7% could be the result of camera angles and the impact of perspective. Disney’s claim appears to be a reasonably good fit to the golden ratio, but let’s keep looking.
Entablature to Column
The entablature, the beam-like structure, lies horizontally above columns and rests on the capitals. Disney shows that the height of the entablature in relation to the width of the capital forms a golden rectangle. The golden rectangles shown in blue on two capitals in the high resolution image below reveal that these dimensions on the Parthenon do in fact embody the golden ratio very accurately. Note that there is a dividing line on the entablature between the frieze on top and the architrave on the bottom. This line is very closely aligned to the golden ratio of the height of the entablature. You’re golden on this one, Donald!
Triglyphs
Disney shows that the fifteen triglyphs within the frieze are also golden rectangles. In the article I wrote on this in 2013, I showed that the triglyphs are in fact golden rectangles in portrait orientation. Beyond that, the triglyph and metopes together create yet another golden ratio in landscape orientation. Donald is golden on this one too.
This is particularly interesting because this illustrates one of the simplest constructions of the golden ratio. Look at this the geometric construction below to see the similarity:
Euclid used the same concept for his first golden ratio construction in Elements. It’s in Book II, Proposition 11 (with the red arc added by me):
Parthenon Floor Plan
My research for my 2018 book on the golden ratio revealed that there are also golden ratios to be found in its floor plan.
Parthenon Golden Ratio Summary
So between the Disney’s “Donald in Mathmagic Land,” my 2013 article and my 2018 book, we find the following golden ratios in the Parthenon:
Facade:
- Facade – Overall height vs width, if you accept starting from the second step. Using that approach, there is then another golden ratio from the bottom of the second step to the top of the columns to the top of the roof.
- Triglyph – Rectangles formed by the Triglyphs.
- Meotope/Triglyph – Rectangles formed by each pair of Meotopes and Triglyphs.
- Entablature/Architrave – Height of the Entablature to the height of the Architrave.
- Frieze/Roofline – Height from the Column top to the top of the Frieze to the height of the roof top.
- Capitals/Entablature – Width of each pair of column Capitals in relation to the height of the Entablature.
Floor Plan:
- Perimeter Outside Columns/Chamber wall – Center points of the exterior columns in relation to the position of the wall between the two interior chambers.
- Perimeter Columns/Inner columns – Center points of the exterior columns in relation to the centers of the four columns in the smaller interior room.
- Building width/Entry width – Width of the inner enclosed structure of the Parthenon in relation to the width of the entry way into the large interior chamber.
A historical timeline perspective
Beyond the physical evidence itself, history tells us that followers of Pythagoras were using the golden ratio-filled pentagram as their secret symbol around 500 BC.
By 300 BC, Euclid had written the first known recorded reference to the golden ratio, defining it and presenting it in numerous geometric constructions in his treatise Elements.
The Parthenon was constructed between these two dates, from 447 to 432 BC.
Quack … or not so quack here?
So what do YOU think? Is this a story that is “without foundation?” Is Donald a “quack?” Or, is there reasonable evidence to conclude that the architects of the Parthenon used the golden ratio in its design?
