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# 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."

Cube, Enneacontahedron, Great Rhombic Triacontahedron, Great Rhombicuboctahedron, Hypercube, Parallelogram, Polar Zonohedron, Rhombic Dodecahedron, Rhombic Icosahedron, Rhombohedron, Rhombus, Zonohedron, Zonotope

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## 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 The Beauty of Geometry: Twelve Essays. New York: Dover, 1999.Eppstein, D. "Zonohedra and Zonotopes." http://www.ics.uci.edu/~eppstein/junkyard/zono/.Eppstein, D. "Ukrainian Easter Egg." http://www.ics.uci.edu/~eppstein/junkyard/ukraine/.Fedorov, E. S. "The Symmetry of Regular Systems of Figures." Zap. Mineralog. Obsc. (2) 28, 1-146, 1891. Reprinted as Symmetry of Crystals. American Crystallographic Assoc., 1971.Fedorov, E. S. "Elements of the Study of Figures." Zap. Mineralog. Obsc. (2) 21, 1-279, 1885. Reprinted Moscow: Izdat. Akad. Nauk SSSR, 1953. http://www.research.att.com/~njas/doc/fedorov.ps.Fedorov, E. S. "Elements of the Theory of Figures." Imp. Acad. Sci., St. Petersburg 1885. Reprinted Moscow: Izdat. Akad. Nauk SSSR, 1953.Fedorov, E. S. Zeitschr. Krystallographie und Mineralogie 21, 689, 1893.Hart, G. "Zonohedra." http://www.georgehart.com/virtual-polyhedra/zonohedra-info.html.Harp, G. W. "Zonohedrification." Mathematica J. 7, 374-383, 1999.Kelly, L. M. and Moser, W. O. J. "On the Number of Ordinary Lines Determined by Points." Canad. J. Math. 1, 210-219, 1958.Towle, R. "Zonohedra." http://personal.neworld.net/~rtowle/Zonohedra/zonohedra.html.Towle, R. "Graphics Gallery: Polar Zonohedra." Mathematica J. 6, 8-12, 1996. http://library.wolfram.com/infocenter/Articles/3335/.

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Equilateral Zonohedron

## Cite this as:

Weisstein, Eric W. "Equilateral Zonohedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EquilateralZonohedron.html