Drag the solid to turn it

There are exactly thirteen. A solid counts if every face is a regular polygon and every vertex looks the same as every other, but not all the faces are alike - that last clause is what rules the Platonic five out and the prisms and antiprisms, which go on forever, are set aside by convention. Archimedes described them in a lost book; Pappus lists them as his, and Kepler rediscovered and named them in 1619.

Nothing here is written down but the five Platonic seeds. Every solid is made from one of them by cutting its corners, taking its edge midpoints, sliding its faces apart, or twisting them; the four that those operations only approximate are then relaxed until every edge touches one sphere. The faces, duals, measurements and nets all follow from the coordinates, so the numbers and the picture cannot disagree.

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(c) 2026 Olivier Giulieri