One day while thumbing through a book I noticed what seemed a deliberate awkwardness in Leonardo da Vinci’s Bacchus/John the Baptist. The pose, pointing with both hands, seemed curiously clumsy for Leonardo, and I had never seen anything clumsy from him before. I remembered that he left illustrations of hexagonal geometry. This geometry is easily drawn with a compass and ruler, without making any alterations in compass angle; the first two points used in its construction determine the rest of the grid, thus the entire grid (which is quite complex) can be indicated with only two points.
So I cut out the picture and went to work. The result was suggestive but far from conclusive, so I started going through art books looking for people pointing. Eventually I found over 300 examples of paintings and sculpture done this way: Rembrandt’s Night Watch, Velasquez’s Meninas, Manet’s Luncheon on the Grass; some of the most famous artworks in the world. This extends in an unbroken line from at least the early Renaissance until today; MC Escher frequently used hexagonal grids in his work, and contemporary artist Odd Nerdrum confirmed my analysis of two of his paintings. Since this is such an extravagant claim, and many readers will no doubt be incredulous, as I was, I will here provide a few quotes supporting it:
Charles Bouleau, The Painter’s Secret Geometry:
In the Middle Ages the ‘geometry’ of a work of art, whether picture, bas-relief or page of manuscript, consisted chiefly in the use of the regular polygons as an armature, as an interior framework, figures that were sometimes quite complicated, with five, six or eight sides, not forgetting the double figures formed by the star pentagons and hexagons.
Albrect Durer, The Painter’s Manual:
Considering, however, that this is the true foundation for all painting, I have proposed myself to propound the elements for the use of all eager students of Art, and to instruct them how they may employ a system of Measurement with Rule and Compass, and thereby learn to recognize the real Truth, seeing it before their eyes.
It is necessary to keep one’s compass in one’s eyes and not in the hand, for the hand executes, but the eye judges.
Dionysius of Fourna, Painter’s Manual:
When you want to draw on a wall, first level the surface and then attach pieces of wood to the legs of a pair of metal compasses, to make them as long as you want, and tie a brush to one end so that you can mark with color the proportions of the figure and describe their halos. When you have marked the proportions of the figure, take some ochre and draw first with a watery solution.
Salvador Dali, Fifty Secrets of Magic Craftsmanship:
I tell you here, young painter, yes, yes, yes, and Yes! you must, especially during your adolescence, make use of the geometric science of guiding lines of symmetry to compose your pictures. I know that painters of more or less romantic tendency claim that these mathematical scaffoldings kill and artist’s inspiration, giving him too much to think and reflect upon. Do not hesitate at that moment to answer them that on the contrary it is in order not to have to think and reflect upon them the you make use of the properties, unique and of a natural magic, derived from the wise use of the golden section, and called the divina proporzione by Luca Pacioli in his memorable book, the most important of aesthetic treatises…
…As I do not wish you to spend days and killing hours which you might devote to painting at your mathematical calculations, I shall now reveal to you the secret of the compass – and this is Secret Number 47 – by means of which you will be able automatically to find as many golden sections as you wish, without having recourse to the painful geometric operation for which you often need an immense compass, requiring that you go beyond the area of your painting, ant this is often so inconvenient that your laziness will counsel you at last to get along without such a proportion…
…And the fact that such compasses are not currently for sale at paint dealers is but the proof of the lack of geometric rigor of schools of art, and of modern painters in particular.