Zonotope (EntityClass, 8)
From Hi.gher. Space
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== Dissection of zonotopes == | == 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. | 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. | ||
| + | |||
| + | == Table of notable zonohedra == | ||
| + | {| style='width: 100%;' | ||
| + | !style='width: 25%; font-weight: bold;'|Zonohedron | ||
| + | !style='width: 25%; font-weight: bold;'|Dual | ||
| + | !style='width: 25%; font-weight: bold;'|Alternation | ||
| + | !style='width: 25%; font-weight: bold;'|Alternation's dual | ||
| + | |- | ||
| + | |[[Cube]]||[[Octahedron]]||[[Tetrahedron]]||[[Tetrahedron]] | ||
| + | |- | ||
| + | |[[Hexagonal prism]]||[[Hexagonal bipyramid]]||[[Octahedron]]||[[Cube]] | ||
| + | |- | ||
| + | |[[Octagonal prism]]||[[Octagonal bipyramid]]||[[Square antiprism]]||[[Tetragonal trapezohedron]] | ||
| + | |- | ||
| + | |[[Decagonal prism]]||[[Decagonal bipyramid]]||[[Pentagonal antiprism]]||[[Pentagonal trapezohedron]] | ||
| + | |- | ||
| + | |[[Octahedral truncate]]||[[Tetrakis hexahedron]]||[[Icosahedron]]||[[Dodecahedron]] | ||
| + | |- | ||
| + | |[[Cuboctahedral truncate]]||[[Disdyakis dodecahedron]]||[[Cubic snub]]||[[Pentagonal icositetrahedron]] | ||
| + | |- | ||
| + | |[[Icosidodecahedral truncate]]||[[Disdyakis triacontahedron]]||[[Dodecahedral snub]]||[[Pentagonal hexecontahedron]] | ||
| + | |- | ||
| + | |[[Rhombic dodecahedron]]||[[Cuboctahedron]]||AYU||AYU | ||
| + | |- | ||
| + | |[[Rhombic triacontahedron]]||[[Icosidodecahedron]]||AYU||AYU | ||
| + | |- | ||
| + | |[[Rhombo-hexagonal dodecahedron]]||[[Square biantiprism]]||AYU||AYU | ||
| + | |- | ||
| + | |[[Rhombic dodecahedral 4-truncate]]||[[Tetrakis cuboctahedron]]||AYU||AYU | ||
| + | |} | ||
== External links == | == External links == | ||
*[http://en.wikipedia.org/wiki/Zonohedron Wikipedia article on Zonohedra] | *[http://en.wikipedia.org/wiki/Zonohedron Wikipedia article on Zonohedra] | ||
*[http://home.inreach.com/rtowle/Polytopes/Chapter2/Polytopes2.html Polytopes: Shadows of Hypercubes] | *[http://home.inreach.com/rtowle/Polytopes/Chapter2/Polytopes2.html Polytopes: Shadows of Hypercubes] | ||
Revision as of 15:06, 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.
