hexahedron as planar graph


Leonardo da Vinci (1452-1519)

Leonardo da Vinci made illustrations of solids for Luca Pacioli's 1509 book The Divine Proportion. These are the first illustrations of polhedra ever in the form of "solid edges", allowing one to see through to the structure. However, it is not clear whether Leonardo invented this new form or whether he was simply drawing from "life" a series of wooden models with solid edges which Pacioli designed. One of these depicted solids is the hexahedron. or cube.

drag the grey points

Drag the grey points of the hexahedron on the left to the graph on the right, so that the vertices and the edges of drawing and graph correspond.
The hexahedron is one of 5 Platonic graphs. These solids have congruent vertices, faces, edges and angles. In the planar draing and the graph you can clearly see that a hexahedron has got 8 vertices, 12 edges and 6 faces. This follows Euler's formula. Euler stated that convex polyhedra, with v the number of vertices, e the number of edges and f the number of faces, always follow the rule v - e + f = 2. For a tetrahedron we get 8 - 12 + 6 = 2.