Platonic Solids

Graph Theory · Axiom Academy

Exploring the five regular polyhedra through the lens of graph theory and planar graphs 1. The Five Platonic Solids as Planar Graphs A Platonic solid is a convex polyhedron where all faces are congruent regular polygons and the same number of faces meet at each vertex. When we project a Platonic solid onto a plane (stereographic projection), we get a planar graph. Tetrahedron: 4 triangular faces, 3 faces meet at each vertex Cube (Hexahedron): 6 square faces, 3 faces meet at each vertex Octahedron: 8 triangular faces, 4 faces meet at each vertex Dodecahedron: 12 pentagonal faces, 3 faces meet at each vertex Icosahedron: 20 triangular faces, 5 faces meet at each vertex 2. Euler's Formula and the Five Platonic Solids For any convex polyhedron (and connected planar graph), Euler's formula states: where V is the number of vertices, E is the number of edges, and F is the number of faces (including the outer infinite face in planar graphs). For a Platonic solid with notation p, q , we can derive: Each face has p edges, and each edge borders 2 faces: pF = 2E Each vertex has q edges, and each edge connects 2 vertices: qV = 2E Substituting into Euler's formula and solving shows that only 5 regular solids exist! 3. Graph Properties of the Platonic Solids Each Platonic solid has distinct graph properties. Let's examine their vertices (V), edges (E), and faces (F): Notice that Euler's formula V - E + F = 2 holds for all five solids! 4. Self-Dual Pairs and Duality

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