Zonohedron#Zonotopes

A zonotope referred to in the geometry, the Minkowski sum of line segments ( the generators of Zonotops ). It is therefore a zonotope in d -dimensional space, if the d- dimensional vectors.

Properties

A zonotope is always a convex polytope, and according to the definition adopted here, the origin is the center of Zonotops. Each zonotope is point-symmetrical to its center. Every facet of a Zonotops is again a zonotope. Zonotope the above is a projection of the k-dimensional unit cube in the d- dimensional space, that is, in matrix notation, the matrix representing a projection of the column with the generators of the unit cube.

Zonoeder

Zonotope in a 3 -dimensional space is referred to as Zonoeder. It is usually assumed that the Zonoeder not limited to one level, the generators so are not coplanar.

Construction of a Zonoeders

Corners, edges, and faces a Zonoeders can be constructed from the generators and then graphically represented for example. The inductive structure is particularly clear: to an already constructed zonotope a new line segment is added. For example, to be added to the already constructed three -dimensional unit cube the segment. For this purpose, the cube is cut along the edges that touch the segment. Then the halves are each shifted by the vector, and, and the resulting gap in the new zone is closed.

Lift halves

Fill gap

Example

The Zonoeder with the generators represents the truncated octahedron

Credentials

• Eppstein, David: Zonohedra and zonotopes. In: Mathematica in Education and Research. 5, No. 4, 1996, pp. 15-21.