A Platonic solid is a convex polyhedron whose faces are identical regular polygons, with the same number of faces meeting at every vertex.

There are five. Not because of mysticism. Because of a counting argument.

At a vertex, at least three faces must meet. Their angles must sum to less than 360°, or the figure flattens into the plane.

• Equilateral triangles (60°): 3, 4, or 5 can meet. Six lie flat.

• Squares (90°): only 3 can meet. Four lie flat.

• Regular pentagons (108°): only 3 can meet.

• Hexagons (120°): three already lie flat. No solid.

Hence tetrahedron, octahedron, icosahedron, cube, dodecahedron.

Plato dressed them as fire, air, water, earth, and the cosmos. Keep the table if it helps you remember. Drop it if it becomes a slogan.

The same joints show up in crystal packing, geodesic domes, and the twelve-faced thinking behind a zodiac wheel. If you want the constructions — vesica, phi, Flower of Life, Kepler’s ellipses, merkaba as a star tetrahedron — they are in Geometry & Astral Connections.

Draw the proof on paper before you decorate it.