Vertex figures: duality and Platonic solids | Tangente

In Euler's formula for convex polyhedra, F – E + V = 2, the number of faces F and the number of vertices V play symmetric roles. This symmetry gives rise to duality. In three-dimensional space, duality is an operation that transforms polygons into points and points into polygons.

But how can we construct a polygon from a vertex of a polyhedron? This is where the vertex figure comes in.

The construction is simple: start at a vertex and move the same distance along every edge incident to it. The concept of a vertex figure allows us to characterize the vertices of a polyhedron in terms of polygons. We can therefore describe a polyhedron using polygons alone: its faces and vertex figures.

This concept even provides a remarkably concise characterization of the Platonic solids: a polyhedron is regular if its faces are congruent regular polygons and its vertex figures are congruent regular polygons. The Schläfli symbol for polyhedra emerges naturally!