Zonotope (EntityClass, 8)

From Hi.gher. Space

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.

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.

Keiji's conjectures

  • Conjecture: No finite, convex zonohedron is self-dual.
Thought process: The minimum number of edges each face of a zonohedron can have is 4, so to be self-dual all vertices would have to have order 4. But this can only happen if it is some rhombic tiling of the plane, thus infinite. As for zonohedra with faces of 6 or more sides, that would fail the theorem that states a planar graph must have at least one vertex of order 5.
  • Stronger conjecture: No finite, convex zonotope of at least three dimensions is self-dual.
  • Even stronger conjecture: Every finite, convex zonotope of n ≥ 3 dimensions contains at least one vertex of order n. Therefore, no such zonotope has a dual which is also a zonotope.

Table of notable zonohedra

# G's Zonohedron Dual Alternation Alternation's dual
4Hexagonal prismHexagonal bipyramidOctahedronCube
5Octagonal prismOctagonal bipyramidSquare antiprismTetragonal trapezohedron
6Decagonal prismDecagonal bipyramidPentagonal antiprismPentagonal trapezohedron
6Octahedral truncateTetrakis hexahedronIcosahedronDodecahedron
8Cuboctahedral truncateDisdyakis dodecahedronCubic snubPentagonal icositetrahedron
15Icosidodecahedral truncateDisdyakis triacontahedronDodecahedral snubPentagonal hexecontahedron
4Rhombic dodecahedronCuboctahedronCube, octahedronOctahedron, cube
6Rhombic triacontahedronIcosidodecahedronDodecahedron, icosahedronIcosahedron, dodecahedron
5Rhombo-hexagonal dodecahedronSquare biantiprismTriaugmented triangular prismStasheff polytope K₅
7Rhombic dodecahedral 4-truncateTetrakis cuboctahedronHexakis truncated tetrahedronTruncated triakis tetrahedron
4Trigonal expanded antitrapezohedron
(as hexagonal prism with surplus facets)
Triangular orthobicupolaEndotrikis antiprismTrigonal trapezosemipyramid
5Tetragonal expanded antitrapezohedron
(as octagonal prism with surplus facets)
Square orthobicupolaEndotetrakis antiprismTetragonal trapezosemipyramid
6Pentagonal expanded antitrapezohedron
(as decagonal prism with surplus facets)
Pentagonal orthobicupolaEndopentakis antiprismPentagonal trapezosemipyramid

External links