Mathematics, Geometry and Art
C.A. Folio 518 recto
Squaring of a Pentagon, a Triangle, an Icosahedron after Plato, Various Other Geometric Studies
pen and ink, traces of black pencil, with incisions and compass point holes, circa 1504-05 or 1513-14
©Veneranda Biblioteca Ambrosiana/Mondadori Portfolio
Folio 518 recto
In contrast to more finished drawings, like the weapons or perspectival exercises, this is a working draft full of different geometric forms with accompanying written notations. The predominance of polyhedrons (a three-dimensional solid with corners, edges and flat surfaces) and inscribed geometric forms reveals that Leonardo was occupied with ongoing geometric problems such as the surface areas of circles and squares (ludo geometrico), the projection of a hemisphere on a plane, and the cubic areas of cones and pyramids. In the centre left of the page is a larger, three dimensional solid, called an icosahedron, consisting of 20 equilateral triangles, one of the five Platonic solids. The presence of Platonic solids links this sheet to the mathematician Luca Pacioli (1447-1517), for whose book on mathematics De divina proportione (1509) Leonardo provided all the illustrations.