03 May 2017

Da Vinci - Homo ad quadratum

The most detailed image of Da Vinci's Vitruvian Man I could find on the internet is this one. (I inquired in Venice about a better one though.) I imported it in Geogebra 5.0, then transformed the image geometrically until the square was as square as possible, with the symmetry (marked by the division in the man's hair) as good as possible. My complete Geogebra file DaVinci.ggb is here at your disposal.

I needed a lot of patient experimenting before I arrived at the above result. If you zoom in deep enough, you'll see that the fit is not perfect. But while you're zoomed in, you could also notice that Leonardo's drawing is not perfect either. Some lines are not exactly where they are said to be, and there is no perfect symmetry between left and right. (BTW, I'll call the arm in the right half the right arm, though it's the man's left.)

To understand an illustration, read the text first. (This simple rule would do miracles in alchemical iconography.) In his text, Da Vinci quotes the Roman architect Vitruvius, who gave simple proportions for the human body. The total height being taken as unit, Da Vinci's proportions, applicable all over the body, are

1, 1/2, 1/4, 1/6, (1/7,) 1/8, 1/10, (1/14) 

where 1/10 is, on one occasion, divided in 3, which amounts to an overall proportion of 1/30. For instance: the arms extended horizontally have length 1, the hand is 1/10, the head (chin to top) 1/8, the face (chin to hairline) 1/10, collarbone to top is 1/6 etc. These numbers are taken from Vitruvius, with the exception of 1/7 (length of the foot, given by Vitruvius as 1/6) and 1/14 (reduction of height if the legs are spread in the appropriate angle). Da Vinci also adds to Vitruvius's list the distance collarbone-hairline, which he gives as 1/7. This number is both redundant and arithmetically wrong. Combining the proportions provided (which we quoted above), one finds this distance to be 1/6-(1/8-1/10), which is 17/120, not 1/7. The latter is a very decent approximation though (indeed, the first convergent in the continued fraction), the difference being 1/840=0.00119…. No doubt Da Vinci was aware that 1/6-1/8+1/10 is not 1/7, so he must have deliberately allowed this tolerance, preferring simple fractions over more complicated numbers (even fractions). This tolerance confirms that, in analysing Da Vinci's Vitruvian, extreme fussiness is unjustified.  

All the proportions required by Da Vinci can be constructed by simple geometry. Halving segments readily provides 1/2, 1/4 and 1/8. The remaining proportions are 1/6, 1/7, 1/10, 1/14 and 1/30. Thanks to Da Vinci's sloppy arithmetic, 1/7 comes for free, and 1/14 is half of this. So, constructing 1/6, 1/10 and 1/30 would do. An 8x8 grid offers plenty of opportunities to do so, and below is one way to proceed. Points on the sides are 1/8 apart. The blue lines and the point A' belong to the grid. The white segment determines B' (1/6 from the top), the yellow one C' (1/10 above the upper blue line), and the green one D' and E', dividing the height 1/10 into three equal parts. Similar right triangles is all one has to look at.

The Homo ad quadratum designed by Vitruvius is composed of simple fractions, and Da Vinci is nothing else but a very accomplished student of the great man. Simple fractions, that's all! (so far)