Equilateral Zonohedron

An equilateral zonohedron is a zonohedron in which the line segments of the star on which it is based are of equal length (Coxeter 1973, p. 29). Plate II (following p. 32 of Coxeter 1973) illustrates some equilateral zonohedra. Equilateral zonohedra can be regarded as three-dimensional projections of -dimensional hypercubes (Ball and Coxeter 1987).

-prisms are zonohedra and may be equilateral. The following table summarizes some equilateral zonohedra together with their basis vectors. As can be seen, a single Platonic solid (the cube), three Archimedean solids (the great rhombicosidodecahedron, great rhombicuboctahedron, and truncated octahedron), and two Archimedean dual (the rhombic dodecahedron and rhombic triacontahedron) are equilateral zonohedra (Ball and Coxeter 1987, Towle 1996).

zonohedron		basis vectors
cube	3	octahedron diameters
great rhombicosidodecahedron	15	icosidodecahedron diameters
great rhombicuboctahedron	9	6 cuboctahedron diameters plus 3 diameters of the octahedron inscribed at the center of its square faces
rhombic dodecahedron	4	cube diameters
rhombic enneacontahedron	10	dodecahedron diameters
rhombic icosahedron	5	pentagonal antiprism diameters
rhombic triacontahedron	6	icosahedron diameters
truncated octahedron	6	cuboctahedron diameters

Regular zonohedra have bands of parallelograms which form equators and are called "zones."