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Zonotope


A zonotope is a set of points in d-dimensional space constructed from vectors v_i by taking the sum of a_iv_i, where each a_i is a scalar between 0 and 1. Different choices of scalars give different points, and the zonotope is the set of all such points. Alternately it can be viewed as a Minkowski sum of line segments connecting the origin to the endpoint of each vector. It is called a zonotope because the faces parallel to each vector form a so-called zone wrapping around the polytope (Eppstein 1996).

A three-dimensional zonotope is called a zonohedron.

There is some confusion in the definition of zonotopes (Eppstein 1996). Wells (1991, pp. 274-275) requires the generating vectors to be in general position (all d-tuples of vectors must span the whole space), so that all the faces of the zonotope are parallelotopes. Others (Bern et al. 1995; Ziegler 1995, pp. 198-208; Eppstein 1996) do not make this restriction. Coxeter (1973) starts with one definition but soon switches to the other.

The combinatorics of the faces of a zonotope are equivalent to those of an arrangement of hyperplanes in a space of one fewer dimension so, for example, zonohedra correspond to planar line arrangements (Eppstein 1996).


See also

Zonohedron

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References

Bern, M.; Eppstein, D.; Guibas, L.; Hershberger, J.; Suri, S.; and Wolter, J. "The Centroid of Points with Approximate Weights." Proc. 3rd Eur. Symp. Algorithms. New York: Springer-Verlag, pp. 460-472, 1995.Coxeter, H. S. M. "Zonohedra." §2.8 in Regular Polytopes, 3rd ed. New York: Dover, pp. 27-30, 1973.Eppstein, D. "Zonohedra and Zonotopes." http://www.ics.uci.edu/~eppstein/junkyard/zono/.Eppstein, D. "Zonohedra and Zonotopes." Mathematica in Educ. Res. 5, 15-21, 1996. http://www.ics.uci.edu/~eppstein/junkyard/ukraine/ukraine.html.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.Ziegler, G. M. Lectures on Polytopes. New York: Springer-Verlag, 1995.

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Zonotope

Cite this as:

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

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