A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated:

• Of the Platonic solids, only the cube is a zonohedron
• Of the Archimedean solids, only the truncated octahedron, the truncated cuboctahedron and the truncated icosidodecahedron are zonohedra.
• Prisms, where the base is a regular polygon with an even number of sides and the sides are squares give an infinite family of vertex-regular zonohedra.

Two other significant zonohedra occur amongst the duals of the Archimedean solids, these being the rhombic dodecahedron and the rhombic triacontahedron. The rhombic enneacontahedron, also is a zonohedron.

Mathematically, the zonohedra can be characterised as being the Minkowski sums of line segments, and this characterisation allows the definition to be generalised to higher dimensions, giving zonotopes.

External links

• Geometry Junkyard Zonohedron page: http://www.ics.uci.edu/~eppstein/junkyard/zono.html