Concept

Symmetry group

The set of all transformations leaving an object where it was found, taken together with composition. Its size measures how symmetric the object is, and its structure distinguishes objects that a count of symmetries alone cannot tell apart.

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

Every turn that leaves a cube where it was. A cube in wireframe beside a table of its rotation axes: 3 of order 4, 4 of order 3, 6 of order 2, totalling 24 turns including the one that does nothing.

The five solids as three groups

There are five regular solids and only three groups of rotations between them, because a solid and its dual share their symmetries exactly. The largest of the three is the smallest group with no way of coming apart, which is why the general equation of the fifth degree has no formula.

geometry · regular polyhedra

Named alongside it

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

Duality Platonic solids Alternating group Archimedean solid Classification Conjugacy class Euler characteristic Group order Permutation Polyhedron Regular polygon Rotation

All concepts