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).
Regular zonohedra have bands of parallelograms
which form equators and are called "zones ."
See also 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/ . Referenced
on Wolfram|Alpha Equilateral Zonohedron
Cite this as:
Weisstein, Eric W. "Equilateral Zonohedron."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/EquilateralZonohedron.html
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