Concept

Archimedean solid

A convex solid whose faces are regular polygons of more than one kind, with every corner surrounded by the same faces in the same order. There are thirteen, apart from the prisms and antiprisms, and eleven of them come from reflecting one point in the mirrors of a regular solid.

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 9 mirrors of the octahedral group, cutting the sphere into 48 triangles. A sphere crossed by the 9 great circles of the octahedral group's mirror planes, dividing it into 48 triangles with one shaded.

Three mirrors make every solid

Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.

geometry · Regular polyhedra

Named alongside it

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

Euler characteristic Platonic solids Classification Duality Group Polyhedron Reflection Regular polygon Sphere Symmetry Symmetry group Truncation

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