Zonotope (EntityClass, 8)
From Hi.gher. Space
(→Table of notable zonohedra: add) |
(→Table of notable zonohedra: add) |
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|[[Icosidodecahedral truncate]]||[[Disdyakis triacontahedron]]||[[Dodecahedral snub]]||[[Pentagonal hexecontahedron]] | |[[Icosidodecahedral truncate]]||[[Disdyakis triacontahedron]]||[[Dodecahedral snub]]||[[Pentagonal hexecontahedron]] | ||
|- | |- | ||
| - | |[[Rhombic dodecahedron]]||[[Cuboctahedron]]|| | + | |[[Rhombic dodecahedron]]||[[Cuboctahedron]]||[[Cube]], [[octahedron]]||Octahedron, cube |
|- | |- | ||
| - | |[[Rhombic triacontahedron]]||[[Icosidodecahedron]]|| | + | |[[Rhombic triacontahedron]]||[[Icosidodecahedron]]||[[Dodecahedron]], [[icosahedron]]||Icosahedron, dodecahedron |
|- | |- | ||
|[[Rhombo-hexagonal dodecahedron]]||[[Square biantiprism]]||[[Triaugmented triangular prism]]||[[Stasheff polytope K₅]] | |[[Rhombo-hexagonal dodecahedron]]||[[Square biantiprism]]||[[Triaugmented triangular prism]]||[[Stasheff polytope K₅]] | ||
|- | |- | ||
| - | |[[Rhombic dodecahedral 4-truncate]]||[[Tetrakis cuboctahedron]]|| | + | |[[Rhombic dodecahedral 4-truncate]]||[[Tetrakis cuboctahedron]]||[[Hexakis truncated tetrahedron]]||[[Truncated triakis tetrahedron]] |
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Revision as of 19:36, 21 November 2011
A zonotope is a polytope which can be constructed as the Minkowski sum of a set of vectors, or line segments with one endpoint at the origin. These vectors are known as the generators of the zonotope.
There are many other equivalent definitions:
- a projection of an n-hypercube, where n is the number of generators;
- a polytope which can be alternated;
- a polytope whose facets are all convex with point symmetry (note that they need not have brick symmetry).
Not all zonotopes are bricks. However, every zonotope can be deformed into a brick with the same topological structure as the original zonotope. On the other hand, it is important to note that this does not work the other way around: there are many bricks (such as the octahedron) which cannot be deformed into a zonotope with the same topological structure as the original brick.
Dissection of zonotopes
One important property of zonotopes is that they can always be dissected into a number of primitive zonotopes. A primitive zonotope is an n-dimensional zonotope with n generators; it follows that all primitive zonotopes are affine transformations of hypercubes.
