Da Vinci - Homo ad circulum

(Previous episode: Da Vinci - Homo ad quadratum)

Inscribing a man in a circle originates with Vitruvius, who famously wrote:

If a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centred at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. 

Da Vinci is more explicit than Vitruvius. In fact, he provides us with everything required to complete the drawing. The one sentence he gives contains the following elements (in his words, though not in that order):

1. if you raise your hands enough that your extended fingers touch the line of the top of your head, know that the centre of the extended limbs will be the navel.
2. open your legs enough that your head is lowered by one-fourteenth of your height. 
3. the space between the legs will be an equilateral triangle. 

For the sole circle, without the displaced legs, (1) suffices. Leonardo does not say what the rotation center for the arms is, but after some experimenting one finds that it must be the point which we coloured white in the image below. It is the intersection of two lines already used for the Homo ad Quadratum: the horizontal line of the arms (1/6 below the top), and the line defining the armpit (1/8 off of the central line). Rotating the arm upward until it meets the upper side of the square (white arc below) we obtain a point of the circle. The man's heel is also on the circle, and the center follows from the perpendicular bisector of the chord. The circle fits to perfection, see image below.

Remarks.

A. What will not work: the golden ratio. It's very easy to obtain the position of the navel using the Golden Ratio. This results in a circle which is way too large, see image below.

B. What will not work: the simple fraction 3/5. Vitruvius commentators invariably place the navel at a height of 3/5. Leonardo did not, see image below. The horizontal line sits at a height of 3/5, and the corresponding circle is way too small.

C. What could work: 3/5+1/160. We constructed the center of the circle geometrically, and no doubt Leonardo did likewise. We can however also compute the radius. Courtesy of Pythagoras, the equation for the radius x is

yielding

This irrational value is very close to the rational number 3/5+1/160, which is 0.60625. Tiny as it is, 1/160 is easily obtained from the Homo ad quadratum. The distance hairline-top is 1/8-1/10=1/40 and it suffices to construct a fourth of it, which is a trivial matter. Thus the circle with radius 3/5+1/160 (i.e. 97/160) can be constructed entirely with readily available rational entities.

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The circle having been drawn, we turn to Leonardo's statement (2) to determine the position of the displaced feet. In the close-up below, the white line is 1/14 above ground level:

Indeed, that's where the displaced feet would be if they were on a horizontal plane, facing forward. (Look at the big toe left.)

We are left with statement (3), that "the space between the legs" will be an equilateral triangle. The image below gives one of the possible interpretations of that statement.

The equilateral triangle in yellow sits on the white line at height 1/14. While it does more or less fill the (ill defined) "space between the legs", the fit is very poor. Thus in the image below (Giovanni Franceso Ferrero 1845, here) the triangolo equilatero is wishful thinking: it is not equilateral!