Concept

Classification

A complete list of the objects of some kind, together with a proof that nothing else qualifies. What distinguishes it from a catalogue is the second half: a classification says what does not exist, which is the part no amount of searching supplies.

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

Also named here as vertex configuration — the same set of essays touches all of them, so they are one junction rather than several.

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.

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

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