Zonotope

In geometry, a zonotope denotes the Minkowski sum of line segments (the generators of the zonotope). So it is a zonotope in d-dimensional space if there are vectors with d entries. ${\displaystyle Z:=[-1,1]v_{1}+...+[-1,1]v_{k}}$ ${\displaystyle v_{1},...,v_{k}}$

properties

A zonotope is always a convex polytope , and according to the definition chosen here, the origin is the center of the zonotope. Each zonotope is point-symmetrical to its center. Each facet of a zonotope is in turn a zonotope. The above zonotope is a projection of the k-dimensional unit cube into the d-dimensional space, i.e. in matrix notation , the matrix with the generators as columns representing a projection and being the unit cube. ${\displaystyle Z:=PQ}$ ${\displaystyle P:={\begin{bmatrix}v_{1}...\,v_{k}\end{bmatrix}}}$ ${\displaystyle Q:={\begin{pmatrix}[-1,1]\\\vdots \\{\begin{bmatrix}-1,1\end{bmatrix}}\end{pmatrix}}}$

Zonahedron

A zonotope in 3-dimensional space is called a zonohedron. It is usually assumed that the zonehedron is not limited to one plane, i.e. that the generators are not coplanar .

Construction of a zonahedron

Corners, edges and facets of a zonahedron can be constructed from the generators and then represented graphically, for example. The inductive construction is particularly clear: a new line segment is added to an already constructed zonotope. For example, the already constructed three-dimensional unit cube to the segment with are added. To do this, the cube is cut open along the edges that are tangent to the segment. Then the halves are shifted by the vector and , and the resulting gap is closed by the new zone . ${\displaystyle Q}$ ${\displaystyle [-1,1]v}$ ${\displaystyle v:=(1\,\,1\,\,1)^{T}}$ ${\displaystyle v}$ ${\displaystyle -v}$

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  Cut the cubes

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  Lift halves

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  Fill the gap

example

The zonahedron with the generators represents the truncated octahedron . ${\displaystyle {\begin{pmatrix}1\\1\\0\end{pmatrix}},{\begin{pmatrix}1\\-1\\0\end{pmatrix}},{\begin{pmatrix}1\\0\\1\end{pmatrix}},{\begin{pmatrix}1\\0\\-1\end{pmatrix}},{\begin{pmatrix}0\\1\\1\end{pmatrix}},{\begin{pmatrix}0\\1\\-1\end{pmatrix}}}$

literature

• Harold Scott MacDonald Coxeter : Regular Polytopes . Dover Publications, New York 1973, ISBN 0-486-61480-8 , pp. 27-30 .
• David Eppstein : Zonohedra and zonotopes . In: Mathematica in Education and Research . 5, No. 4, 1996, pp. 15-21.

Web links

• The Geometry Junkyard: Zonohedra and Zonotopes on ics.uci.edu
• Online Zonotope Builder and Viewer at decatur.de