Zonotope (EntityClass, 8)
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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. | 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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| + | Similarly to many other classes of polytopes, the facets (of any dimension) of a zonotope are also zonotopes themselves. | ||
== Dissection of zonotopes == | == Dissection of zonotopes == | ||
Revision as of 21:03, 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.
Similarly to many other classes of polytopes, the facets (of any dimension) of a zonotope are also zonotopes themselves.
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.
