square pyramid 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 square pyramid.

drag the grey points

Drag the grey points of the pyramid on the left to the graph on the right, so that to top of the pyramid is the central point of the graph.
Platonic graphs have congruent vertices, faces, edges and angles. So the quare pyramid is not a platonic solid, since the square base doesn't correspond with the other 4 triangular faces. In the planar drawing and the graph you can clearly see that a square pyramid has got 5 vertices, 8 edges and 5 faces. This doesn't 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 square pyramid we get 5 - 8 + 5 2. Eulers formula doesn't match for all solids, but only for polyhedra.