Realization (vertex mapping)

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There are multiple ways to realize an abstract polytope in some space. Unfortunately, many of them share the name realization. Unless otherwise specified, this article deals with the most general of these notions.

The idea of a realization is that only the vertices of an abstract polytope need to be placed in space in order to determine the placement of all the other elements. Thus, a realization is nothing more than a function from the vertex set into some other set , the space, whose members are called points.[1] The image of is called the vertex set of the realization. Most often the space will be Euclidean space, or a Riemannian manifold like spherical or hyperbolic space, so that further notions like symmetries or planarity can be defined. In theory however, could be any kind of set.

It's important to remember that realizations carry no intrinsic notion of an interior, nor of the interior of any of its elements. In illustrations, edges of realizations are often drawn as solid segments, but this is but a visual aid.

This is a quite general definition, which makes it very broadly applicable. For instance, this is the way in which skew polytopes can be said to exist in Euclidean space. Conversely, this definition can be quite loose, and many further restrictions, like planarity or faithfulness often need to be imposed.

Faithful realizations[edit | edit source]

Given the function , further functions can be defined inductively. Let be the vertex set of the realization, and define as the power set of . Then the function [note 1] is defined through

For example, the function maps each edge to a set comprising its two endpoints. If a quadrilateral face has vertices located at points in cyclic order, maps that face to .

If each is bijective, the realization is said to be faithful. Otherwise, the realization is said to be degenerate. If is bijective, the realization is called vertex-faithful.

Every polytope admits a realization where all vertices are mapped to the same point. This is called the trivial realization. Unless the polytope in question is either a nullitope or a point, this realization will be degenerate.

Some polytopes, such as the digon, admit no faithful realizations. This happens when two (j + 1)-faces are adjacent to the exact same set of j-faces. In the case of the digon, both edges contain the exact same set of vertices. By contrast, any other polygon admits a faithful realization, as long as has at least as many points as the polygon has vertices.

Symmetric realizations[edit | edit source]

PointIcosahedronGreat icosahedronHemiicosahedronSkew icosahedron
Symmetric realizations of {3,5}. Click on a node to be taken to the page for that realization.

Every automorphism of an abstract polytope induces a permutation on its vertices. If the polytope is then realized, these automorphisms further induce permutations on the vertex set. If each of these extends to an isometry of the ambient space, then the realization is said to be symmetric.

These isometries induced by the automorphisms of are its symmetries, and form its symmetry group.

Simplex realizations[edit | edit source]

The petrial tetrahedron is the simplex realization of the hemicube.

Every finite abstract polytope that has a symmetric realization, is also symmetrically realized with the vertices of a simplex. For example, the pentagon has a usual realization as a planar polygon, thus it also has a realization with the vertices of a 4-simplex, the pentagonal-pentagrammic coil. The hemioctahedron has three vertices, so its simplex realization would be on the 2-simplex, however this realization is degenerate with all of its faces coinciding, thus the hemioctahedron has no non-degenerate symmetric realizations.

Abstract polytopes will generally have a non-degenerate simplex realization. The simplex realization is only degenerate if the abstract polytope is degenerate to begin with (e.g. dihedra) or if its vertex count is less than or equal to its rank (e.g. hemioctahedron).

Properties[edit | edit source]

  • When the abstract polytope is not a simplex the simplex realization is skew.
  • A simplex realization of dimension r  can be expressed as a blend of r  or fewer components.[2]
  • If a faithful symmetric realization of 𝓟 is pure, then it is similar to a component of the simplex realization of 𝓟.[3]
  • If the simplex realization is pure, it is the only faithful symmetric realization.
  • The simplex realization is always the embedding of the polytope with the highest dimension.
  • If a finite regular polytope doesn't have a non-degenerate simplex realization, it is flat.

Cross-polytope realizations[edit | edit source]

If an abstract polytope has a central involutionary automorphism, it has a symmetric realization on the vertices of a cross-polytope. The dimension of this realization is half the number of vertices of the abstract polytope.

Properties[edit | edit source]

  • If a polytope, 𝓟, has a cross-polytope realization, P , then every pure symemtric realization is either a component of P  or of the simplex realization of the hemi-polytope, P /2.[4] In other words the simplex realization of 𝓟 is exactly P #P /2

Notes[edit | edit source]

  1. indicates the set of i -elements of 𝓟.

References[edit | edit source]

  1. McMullen & Schulte (2002:121)
  2. McMullen & Schulte (2002:129)
  3. McMullen & Schulte (2002:129)
  4. McMullen & Schulte (2002:135)

Bibliography[edit | edit source]

  • McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes. Cambridge University Press. ISBN 0-521-81496-0.