# Truncated cuboctahedron

Truncated cuboctahedron
Truncated cuboctahedron
Click on picture for large version.
Type Archimedean
Faces 12 squares
8 hexagons
6 octagons
Edges 72
Vertices 48
Vertex configuration 4,6,8
Symmetry group octahedral (Oh)
Dual polyhedron disdyakis dodecahedron
Properties convex, semi-regular (vertex-uniform), zonohedron

The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. It has 12 regular square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron.

Note that the name truncated cuboctahedron may be a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get an actual regular truncated cuboctahedron: some of the faces will be irregular polygons. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular. The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.

## Canonical coordinates

Canonical coordinates for the vertices of a truncated cuboctahedron centered at the origin are all permutations of (±1, ±(1+√2), ±(1+√8)).