06 May 2017

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. 
What Vitruvius writes is quite possible, but what Da Vinci shows is an illusion. Geometrically speaking, it is impossible that the heels of the vertical legs and those of the spread legs are on the same circle—unless your legs meet at your navel.

If your legs meet at a point M below your navel N (the exact positions don't matter here), and you move your heel from position H to H', then by the triangle inequality NH' < NM+MH'. If you managed to move your heel along a circle with center N, then NH'=NH. Hence, NH < NM+MH', i.e., NM+MH < NM+MH', which simplifies to MH < MH'. Your leg has miraculously grown, from length L to a bigger length L'. Given this geometrical fact, we must be prepared for some slight cheating.

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.

To see how nicely the center coincides with the navel, compare these two close-ups: left the raw drawing, right with the center of the circle added as a red dot. If you look very carefully, you'll find a dot left by Da Vinci right there.

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


This irrational value is very close to the rational number 3/5+1/160, which is 0.60625. So with Da Vinci the navel sits some 1/160 higher than with the other Vitruvius commentators. The most accomplished of the latter is Claude Perrault (1684). Below is his Homo ad quadratum; the red lines are ours. In the left column you can check the height of the navel: mark 3 in Cinq Parties (five parts) and mark 6 in Dix Parties (Ten parts).

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. Vitruvius would have been pleased! Below is what you get; the twin lines are 1/160 apart. Convincing, no? In my Geogebra view, no difference is observable between the circle with the rational radius and the irrational one.

Summary for the vertical measures:

The circle, one way or another, having been drawn, we turn to (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 observation (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. It does more or less fill "the space between the legs", which is determined by the thickest parts of the legs. (Look at the thigh left.)

The craftsmanship displayed by Da Vinci in this drawing is baffling. What an artist!

P.S.1 Common lore has it that the navel in Da Vinci's man is defined by the golden ratio. This claim is easily refuted by any distortion free image (admittedly hard to come by). In the image below we constructed the golden navel GN and the corresponding circle (violet). Way too big! Neither is the navel at height 3/5. We constructed this "simple navel" SN as well, using the 6/5 slope of the segment CN. This time the circle (green) is too small.

P.S.2 For those secretly thinking of squaring the circle: the area of Da Vinci's circle is 1.1545..., some 15% bigger than the square.