Zonotope (EntityClass, 8)

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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:

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.

Table of notable zonohedra

Zonohedron Dual Alternation Alternation's dual
CubeOctahedronTetrahedronTetrahedron
Hexagonal prismHexagonal bipyramidOctahedronCube
Octagonal prismOctagonal bipyramidSquare antiprismTetragonal trapezohedron
Decagonal prismDecagonal bipyramidPentagonal antiprismPentagonal trapezohedron
Octahedral truncateTetrakis hexahedronIcosahedronDodecahedron
Cuboctahedral truncateDisdyakis dodecahedronCubic snubPentagonal icositetrahedron
Icosidodecahedral truncateDisdyakis triacontahedronDodecahedral snubPentagonal hexecontahedron
Rhombic dodecahedronCuboctahedronAYUAYU
Rhombic triacontahedronIcosidodecahedronAYUAYU
Rhombo-hexagonal dodecahedronSquare biantiprismAYUAYU
Rhombic dodecahedral 4-truncateTetrakis cuboctahedronAYUAYU

External links