The 5 Platonic solids, the 13 Archimedean solids and the 13 Catalan solids dual to them. Each one on this page is built in your browser from a Platonic solid by cutting, twisting or swapping faces for corners, then relaxed until every face is flat.
Cut the corners off a cube with flat planes and push the planes steadily inward. The named solids are just a few stops along the way. Every position in between is a solid too, with the same shape of faces but irregular sides.
A Platonic solid has one kind of face, a regular polygon, and the same number of faces at every corner. There are exactly five, and the proof is how Euclid's Elements ends.
An Archimedean solid allows several kinds of regular face, but every corner has to look the same: some symmetry of the solid carries any corner to any other. Leave out the two endless families of prisms and antiprisms and there are thirteen.
Archimedes wrote about them, but the work is lost. We know it existed because Pappus of Alexandria, more than five centuries later, describes it and lists the thirteen by their faces. Renaissance artists redrew some of them. Kepler worked out the full set again in Harmonices Mundi (1619), argued that there are no others, and gave most of them the names still used: cuboctahedron, icosidodecahedron, truncated cube, snub cube.
A Catalan solid is an Archimedean solid turned inside out: one face for each corner, one corner for each face. Now every face is the same instead of every corner. Eugène Catalan described them in 1865.
| Solid | Faces | V | E | F | V − E + F |
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