30 July 2016

Debunking golden ratio navels (1)

In his book The music of the primes Oxford professor of mathematics Marcus du Sautoy nicely collects in a single paragraph most of the golden ratio myths. Thus (p.27): 
Experiment reveals that a person's height when compared to the distance from their feet to their belly button favours the same ratio [i.e. the golden ratio].
It is not clear what 'favours' means, but it has a statistical flavour. So, let's investigate the (statistical) claim
N/L = φ

where L is body length, N the height of the navel, and φ=(√5-1)/2, approximately 0,61803... Sure enough, in the 19th century, when golden numberism was invented, the navel of "Man" was placed exactly there, but this proves nothing at all. Before that golden era, nobody seemed to have been aware of any role of φ outside mathematics, in human bodies no more than in architecture.

In his Vitruvian Man of 1487, da Vinci placed the navel at 6/10 + 1/160, but he was cheating anyway (verify it here). The general rule with Vitruvian commentators is: 6/10. Thus Cesariano in 1521 (verify it here), and likewise in 1572 (below, the whole book here).

A century later, in 1678, Samuel van Hoogstraten did so for both genders:

Van Hoogstraten even seems to have done some homework on the female body, because her legs are 7/15 long, while his are the good old Vitruvian 7,5/15. Navels remain unisex at 6/10 though.

Our last example (more could be given) is from Perrault's 1684 Vitruvius translation. The blue lines are ours and show that, again, the navel is exactly at 6/10 (gauge left, in Dix Parties, i.e. ten parts).


Dürer was the first to decidely leave Vitruvian scholastics and enter real life, populated with non-Vitruvian men and women. His Four books on human proportion (1528, the entire book here, in German) are entirely devoted to he subject. We will restrict ourselves to the first book, because there the proportions are expressed in fractions of the body height, while in the remainder he switched to a less transparent system. The first book, then, contains detailed views of eleven persons, 5 men, 5 women and 1 young child. He calls the men A,B,C,D,E (ordered from fat to thin), the women A.i, B.i,C.i,D.i,E.i (also from fat to thin) and the child F. Of each person three views are provided: front, sideways and back, all marked with proportions. The vertical proportions are on the back views. They are given as fractions, e.g. 2/11, of the body height.  Numbers that are not recognisable as fractions are to be understood as denominators of fractions with numerator 1. Thus, '3' stands for '1/3'. Sometimes, fractions are to be added. For instance, '10 et 11' stands for '1/10 + 1/11'. Here is the part of man A that is relevant for our purposes.

"In der weichen" means "in the waist", and below it we find "Im nabel", "in the navel". Following Dürer from the top down, we find 1/10 + 1/11, then 2/11 and finally 1/40, adding up to 35/88. This means the height of the navel is 53/88, being 0,6022... , above ground level. We repeated this for the other 10 persons, and obtained the following table, in which we included the averages for the five men and the five women.

Where is the golden ratio? Dürer and da Vinci were perfectly aware of φ, because Pacioli's (entirely mathematical) book on the subject dates from 1498. Had they wanted, they could have used a simple geometrical construction to locate navels at height φ. As neither did, they must not have seen the point of it. Instead, Dürer uses a system in which the golden ratio cannot possibly show up, because φ cannot be written as a fraction. But perhaps golden numberists would settle for somewhat less, like one of the fractions constructed from Fibonacci's sequence

1, 1, 2, 3, 5, 8, 13, ...

 The terms divided by their respective predecessor give

1, 1/2, 2/3, 3/5, 5/8, 8/13, ...
and it is well known that this sequence converges (though not very fast) to φ. Nope, no fraction of the sort to be seen anywhere. In short: no golden ratio anywhere.

There is some doubt as to whether Dürer provides us with real measurements of real individuals, or artistically merged a bunch of data into a single 'average' person of a certain body type. (Read all about it here.) One reason for suspicion is that his proportions are given with exaggerated precision (for which, see next post). Anyhow, a sample of 11 is not statistically meaningful. But, it might not be accidental that the highest navel is found in the thinnest man, and the lowest one in the fattest woman. (Unless this reflects what Dürer was convinced a priori to be the case. The man has been caught on other occasions, see here.) 

For what it's worth: the overall crude impression from Dürer's proportions is, that navels of adults are situated slightly above 0,6 of their body height, higher in men than in women, and higher in thin persons than in fat ones. If the single child is taken into account, one could add: higher with age. [Italics not entirely justified.]

In his sample of ten, Dürer considers people whose height is 7, 8, 9 and then 10 times their head. Later theorists divide an 8-header vertically in 8 parts: top to chin, chin to nipples, nipples to navel and so on. In this scheme, the navel is placed 3/8 from the top, hence on height 5/8, which is 0.625. Among those who explicitely state this rule we find Samuel van Hoogstraten, who frames it in a didactical little poem (High School of Painting, p.57). Yet his 7,5-header (shown above) had a navel height of 3/5. So yes, it would seem that taller people were supposed to have higher navels.

In view of what arm length has revealed (here), it is not unreasonable to expect that statistical navel heights, duely measured, would turn out to depend on sex, race and age. At least, Dürer's 'findings' do not contradict that guess.


(continued here)