Zonotope (EntityClass, 8)
From Hi.gher. Space
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<[#ontology [kind class] [cats Polytope]]> | <[#ontology [kind class] [cats Polytope]]> | ||
| - | + | A '''zonotope''' is a [[polytope]] which can be constructed as the [[Minkowski sum]] of a set of [[vector]]s, or [[line segment]]s with one endpoint at the origin. These vectors are known as the ''generators'' of the zonotope. | |
| - | A '''zonotope''' is a [[polytope]] which can be constructed as the [[Minkowski sum]] of a set of [[vector]]s, or [[line segment]]s with one endpoint at the origin. These vectors are known as the | + | |
There are many other equivalent definitions: | There are many other equivalent definitions: | ||
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*a polytope which can be [[alternated]]; | *a polytope which can be [[alternated]]; | ||
*a polytope whose facets are all [[convex]] with [[point symmetry]] (note that they need not have [[brick symmetry]]). | *a polytope whose facets are all [[convex]] with [[point symmetry]] (note that they need not have [[brick symmetry]]). | ||
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| + | Not all zonotopes are bricks. However, every zonotope can be [[deform]]ed 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. | ||
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| + | == Dissection of zonotopes == | ||
| + | One important property of zonotopes is that they can always be [[dissect]]ed 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 transformation]]s of hypercubes. | ||
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| + | == External links == | ||
| + | *[http://en.wikipedia.org/wiki/Zonohedron Wikipedia article on Zonohedra] | ||
| + | *[http://home.inreach.com/rtowle/Polytopes/Chapter2/Polytopes2.html Polytopes: Shadows of Hypercubes] | ||
Revision as of 14:33, 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.
