Zonohedron

A convex Polyhedron whose faces are Parallel-sided $2m$-gons. There exist $n(n-1)$ Parallelograms in a nonsingular zonohedron, where $n$ is the number of different directions in which Edges occur (Ball and Coxeter 1987, pp. 141-144). Zonohedra include the Cube, Enneacontahedron, Great Rhombic Triacontahedron, Medial Rhombic Triacontahedron, Rhombic Dodecahedron, Rhombic Icosahedron, Rhombic Triacontahedron, Rhombohedron, and Truncated Cuboctahedron, as well as the entire class of Parallelepipeds.

Regular zonohedra have bands of Parallelograms which form equators and are called ``Zones.'' Every convex polyhedron bounded solely by Parallelograms is a zonohedron (Coxeter 1973, p. 27). Plate II (following p. 32 of Coxeter 1973) illustrates some equilateral zonohedra. Equilateral zonohedra can be regarded as 3-dimensional projections of $n$-D Hypercubes (Ball and Coxeter 1987).

See also Hypercube

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 141-144, 1987.

Coxeter, H. S. M. ``Zonohedra.'' §2.8 in Regular Polytopes, 3rd ed. New York: Dover, pp. 27-30, 1973.

Coxeter, H. S. M. Ch. 4 in Twelve Geometric Essays. Carbondale, IL: Southern Illinois University Press, 1968.

Eppstein, D. ``Ukrainian Easter Egg.'' http://www.ics.uci.edu/~eppstein/junkyard/ukraine/.

Fedorov, E. S. Zeitschr. Krystallographie und Mineralogie 21, 689, 1893.

Fedorov, E.W. Nachala Ucheniya o Figurakh. Leningrad, 1953.

Hart, G. W. ``Zonohedra.'' http://www.li.net/~george/virtual-polyhedra/zonohedra-info.html.