10 August 2016

Debunking golden ratio architecture (2)

This stop on the Golden Tour is casual and relaxing. You might say funny, because it doesn't get much sillier than this.

Today's claim is, that the façade of the Parthenon fits in a golden rectangle. For proof, one is usually served a front view of the building (with the upper triangle, which is all but missing, completed), with some golden rectangle as overlay. Most of the time, the fit is poor and lines are thick. But, every now and then, some details are provided. The picture below is accompanied by the specification 'that the bottom of the golden rectangle should align with the bottom of the second step into the structure and that the top should align with a peak of the roof that is projected by the remaining sections.'

Even if this claim were true, and even if the golden rectangle had any esthetic value (quod non), no meaningful impression whatsoever can be expected from a view consisting of a triangular roof, a rectangle, and two steps of four that are visible. The building was not even designed to be viewed from there, and the upper step is—by design—not even straight! To disprove the claim, it suffices to obtain reliable measurements of the Parthenon, including details such as the height of the steps. This is what one gets for the horizontal and the vertical dimensions of the proposed rectangle:

h = 30.604 m
v = 18.162 m (upper step to top of triangle) + 0.552 m (upper step) + 0.512 (second step) = 19.226 m

giving a ratio of v/h=0.62821... Wow! Weren't we expecting φ=0.61803...? And to say that this rectangle, no doubt, was selected among many other possible ones because it was the most 'golden'. It's 'approximately φ' all right, but everything is 'approximately φ' within a certain precision.

A secondary claim, often accompanying the primary one, is that the building abounds with (almost) golden rectangles. Here, we need not spend any time verifying, because the claim means nothing. There are so many rectangles to choose from that doubtlessly many of them will be 'approximately golden'. Here a fine selection offered for your entertainment.

When you're done, you might hunt throughout the Parthenon for rectangles displaying a 9:4 proportion. This time, your findings may even be meaningful, because this simple proportion did play a role in the design.

Ah, lest I forget! I owe you a reliable diagram with Parthenon measurements. You can't beat Athanasios G. Angelopoulos, author of Metron ariston (unfortunately in Greek), who provides us with measurements up to a mm. Here two detailed views, front:

 and sideways:

Measures are in meters, but the additions in bold refer to a different unit of some 45 cm. Wherever possible, the author expresses his 'bold' measurements as mathematical expressions involving roots, powers, φ (which is (√5+1)/2, what we denote by Φ), π and e. You could do the same (describe real numbers as values of special functions, that is, not measure the Parthenon) by consulting Borwein's A dictionary of real numbers.

Good hunting!


Added April 22, 2017. The inventor of Golden Numberism is the 19th century German teacher and writer Adolf Zeising. He says so himself, because the title of his 1854 influential classic translates as
A new theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole of nature and art. 
His weird and sometimes hilarious convictions (deeply rooted in Hegel's idealism) will be discussed separately. Here we restrict ourselves to the two pages (these) where Zeising introduces the Parthenon as the Golden Ratio crown jewel it has remained until today. In his image below we have transferred the vertical measure in red to its horizontal origin.

Zeising claims —the first one to do so— that "the" height (blue) and "the" width (red) of the Parthenon are in a golden proportion, and he provides us with the measures: height=65 German feet of some kind, width=107 feet. The measurements being up to one foot, we only know that the width is from 64.5 to 65.5 feet, and the height from 106.5 to 107.5. Hence, the ratio is between 64.5/107.5=0.600... and 65.5/106.5=0.615... Thanks, Adolf! Whatever the Parthenon ratio is by accident or intention, it's definitely not the golden ratio, because 0.618.... is outside the bounds obtained. Yet Zeising sees only an unimportant fraction of difference between this measurement and his ideal. In spite of the ridiculous number of digits in his models, he's less fussy when reality joins in. On p.310 he considers 11/18 (which is a lousy 0.6111...) as a reasonable attempt of creative nature to realise what he calls our law.

The actual measures given by Angelopoulos are: width=30.604m, height=19.738m if (as in Zeising's drawing) only three steps are taken into account, or 20.038m counting also the fourth step. The ratio is 0.644... resp. 0.654..., both way too big to be golden.