dodecahedron 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 dodecahedron with 12 regular pentagons as faces.

drag the grey points

Drag the grey points of the tetrahedron on the left to the graph on the right, so that the corresponding points match.
The dodecahedron is one of 5 Platonic graphs. These solids have congruent vertices, faces, edges and angles. In the planar drawing and the graph you can clearly see that a dodecahedron has got 20 vertices, 30 edges and 12 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 dodecahedron we get 20 - 30 + 12 = 2.