Concept

Vertex configuration

The list of face sizes met in order when walking round a single corner of a polyhedron. It is what a solid with every corner alike is classified by, and the enumeration of which such lists can close up is how the Archimedean solids are counted.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Solids with every corner alike and more than one kind of face. truncated tetrahedron, cuboctahedron, truncated cube, icosidodecahedron, each cut from a Platonic solid and drawn in projection; every edge in each is the same length and every vertex is surrounded by the same faces.

Thirteen more when one word is dropped

The list of regular solids stops at five because the definition asks for two things at once. Ask for only the second — every corner alike — and thirteen more appear, each of them cut off a Platonic solid at a depth found rather than chosen.

geometry · regular polyhedra

The regular solids of four dimensions. 5-cell, tesseract, 16-cell, 24-cell, each turned in four dimensions and projected to the page; the edges are the pairs of vertices at the shortest distance apart.

Six in four dimensions, and three forever after

The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.

geometry · regular polyhedra

Named alongside it

The objects these essays reach for when they reach for this one.

Classification Duality Platonic solids Polyhedron Archimedean solid Dimension Euler characteristic Polytope Projection Regular polygon Schlafli symbol Simplex

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