Platonic solids are the only convex polyhedra with identical regular faces and the same number of faces at every vertex — perfect symmetry in 3D.
Each obeys Euler’s formula: V – E + F = 2, representing the balance of vertices, edges, and faces in stable geometric chambers.
Tetrahedron — 4 triangular faces
Cube (Hexahedron) — 6 square faces
Octahedron — 8 triangular faces
Dodecahedron — 12 pentagonal faces
Icosahedron — 20 triangular faces
Every solid has a dual where faces and vertices swap roles — e.g., Cube ↔ Octahedron.
These solids’ symmetry groups describe how rotational and reflection symmetry govern stable flow and distribution of forces, like balanced hydraulic networks.
Platonic Solids Explainer
Platonic solids are the only five convex polyhedra with identical regular faces and the same number of faces meeting at each vertex. They are the cornerstone of geometric symmetry.
Each solid obeys Euler’s formula: V – E + F = 2, where V is vertices, E is edges, and F is faces.
Each solid has a dual: Cube ↔ Octahedron, Dodecahedron ↔ Icosahedron, Tetrahedron ↔ Tetrahedron (self-dual).