Zonotope (EntityClass, 8)
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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. | ||
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| + | == 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 at least three dimensions contains at least one vertex of order 3. Therefore, no such zonotope has a dual which is also a zonotope. | ||
== Table of notable zonohedra == | == Table of notable zonohedra == | ||
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|[[Rhombic dodecahedral 4-truncate]]||[[Tetrakis cuboctahedron]]||[[Hexakis truncated tetrahedron]]||[[Truncated triakis tetrahedron]] | |[[Rhombic dodecahedral 4-truncate]]||[[Tetrakis cuboctahedron]]||[[Hexakis truncated tetrahedron]]||[[Truncated triakis tetrahedron]] | ||
| + | |- | ||
| + | |[[Tetragonal expanded antitrapezohedron]]<br />(when deformed into a cube with surplus facets)||[[Octagonal semibipyramid]]||[[Endotetrakis antiprism]]||[[Tetragonal trapezosemipyramid]] | ||
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Revision as of 21:10, 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.
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 at least three dimensions contains at least one vertex of order 3. Therefore, no such zonotope has a dual which is also a zonotope.
