A zonotope is a polytopeMathworldPlanetmath which can be obtained as the Minkowski sumPlanetmathPlanetmath (http://planetmath.org/MinkowskiSum3) of finitely many closed line segmentsPlanetmathPlanetmath in n. Three-dimensional zonotopes are also sometimes called zonohedra. Zonotopes are dual to finite hyperplane arrangements. They are centrally symmetricMathworldPlanetmath, compact, convex sets.

For example, the unit n-cube is the Minkowski sum of the line segments from the origin to the standard unit vectors ei for 1in. A hexagonMathworldPlanetmath is also a zonotope; for example, the Minkowski sum of the line segments based at the origin with endpointsMathworldPlanetmath at (1,0), (1,1), and (0,1) is a hexagon. In fact, any projection of an n-cube is a zonotope.

The prism of a zonotope is always a zonotope, but the pyramidMathworldPlanetmath of a zonotope need not be. In particular, the n-simplex (http://planetmath.org/HomologyTopologicalSpace) is only a zonotope for n1.


  • 1 Billera, L., R. Ehrenborg, and M. Readdy, The cd-index of zonotopes and arrangements, in Mathematical essays in honor of Gian-Carlo Rota, (B. E. Sagan and R. P. Stanley, eds.), Birkhäuser, Boston, 1998, pp. 23–40.
  • 2 Ziegler, G., Lectures on polytopes, Springer-Verlag, 1997.
Title zonotope
Canonical name Zonotope
Date of creation 2013-03-22 15:47:20
Last modified on 2013-03-22 15:47:20
Owner mps (409)
Last modified by mps (409)
Numerical id 7
Author mps (409)
Entry type Definition
Classification msc 52B99
Synonym zonohedron
Synonym zonohedra
Related topic HyperplaneArrangement