Skew polytope

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Skew polytopes have several competing definitions, but the classic example is a polytope of rank n  realized in Euclidean space, but whose vertices do not all lie in an n -dimensional subspace. For example, a skew polygon in general position is realized in Euclidean 3-space or higher, but does not lie in a plane.

Skew polytopes generally have no well-defined interior.

Definitions[edit | edit source]

This section will present several potential definitions of skewness and a comparison of them. Some of the definitions here are not in wide use but are included for comparison purposes.

  1. A polytope is skew when its vertices do not all lie in an n -dimensional subspace, where n  is the rank of the polytope.
  2. A polytope is skew when its vertices do not all lie in an n -dimensional subspace, where n  is the rank of the polytope, or has proper elements which are skew. Equivalently, the polytope is realized in Euclidean space but is not planar.
  3. A polytope is skew when its vertices do not all lie in an n -dimensional subspace, where n  is the rank of the polytope, or has proper elements or figures which are skew.
  4. Given the following two definitions of a polytope, a polytope is skew when it meets definition 2 but not definition 1.
    1. A polytope is an abstract polytope and an injective mapping π  of its elements onto closed connected subsets of Euclidean space, such that
      • for an element x  of rank n , is a set whose affine span has n  dimensions,
      • the boundary of is the union of π  applied to the proper elements of x ,
      • iff .
    2. A polytope is a faithful realization of an abstract polytope in Euclidean space.
  5. A polytope is skew when there exists an abstractly equivalent polytope with the same symmetry whose vertices lie in a lower dimensional subspace than the original polytope.
  6. A polytope is skew when its vertices all lie on an n -sphere or n -horosphere, where n+1  is the rank of the polytope, or has proper elements which are skew.
  7. A polytope of rank r+1  is skew when it does not have a dimension vector of the form .
Comparison of definitions of skew using example polytopes
Polytope Definition
1 2 3 4 5 6 7
Skew cube Skew Skew Skew Skew Skew Skew Skew
Square duocomb Skew Skew Skew Skew Non-skew Skew Skew
Blended octahedron Skew Skew Skew Skew Non-skew Skew Skew
Petrial cube Non-skew Skew Skew Skew Non-skew Skew Skew
Mucube Non-skew Non-skew Skew Skew Non-skew Skew Skew
Regular zigzag Non-skew Non-skew Non-skew Skew Skew Skew Skew
Mucubic dichoron Non-skew Non-Skew Skew Non-skew Non-skew Skew Skew
P tetracube Non-skew Non-skew Skew Non-skew Non-skew Skew Undefined
Z4 Non-skew Non-skew Non-skew Non-skew Non-skew Skew Undefined
Digon Non-skew Non-skew Non-skew Non-skew Non-skew Non-skew Non-skew
Square Non-skew Non-skew Non-skew Non-skew Non-skew Non-skew Non-skew

Some of these definitions are context-specific, and either fail to be meaningful or produce strange results when considered outside the intended context.

  • Definition 3 is generally applied to isogonal polytopes, although there are other classes of polytopes for which it may produce sensible results.
  • Definition 6 is only useful in the context of isogonal polytopes.
  • Definition 7 is only defined for regular polytopes.

Definition 3 is the most widely cited definition for skew polytopes, although it fails to include some polytopes like the zigzag which are commonly considered skew.

Polytopes that meet definition 5 are the results of blending and often have skew in their name, although not all blends meet definition 5.

Skew polygons[edit | edit source]

A skew decagon found in the zigzagging edges around the middle of a pentagonal antiprism.

Consider a 3-dimensional antiprism (more specifically, the alternation of a uniform or semi-uniform prism). The edges can be divided into three sets, those being the edges of the two capping polygons and the zigzagging edges that form a ring around the middle. This last set forms a polygon that is both isogonal and isotopic, making it a regular polygon. The vertices lie in two planes, as opposed to one as seen in conventional polygons.

Uniform polygons[edit | edit source]

All uniform non-skew polygons are regular, but this does not apply to skew polygons. There are skew polygons that are uniform but not regular. Some examples include truncations of regular blended polygons (sometimes called 'wavy polygons'), or the Petrie polygons (see below) of apeirogon - polygon duocombs.

Petrie polygons[edit | edit source]

In the above description, the distance between alternating vertices is the edge length, but this need not be so. In particular, every regular polytope (of 3 or more dimensions) has a corresponding skew polygon. This is formed by taking connecting edges of the polytope, with the following rule: every n – 1 connected edges are in the same facet, but n connected edges do not share a facet.

These are the Petrie polygons.

Skew apeirogons[edit | edit source]

Apeirogons can be considered skew even if they are coplanar. For example, the regular zigzag skew apeirogons formed by the triangle-triangle edges of the apeirogonal antiprism. There are also helical apeirogons, which exist in three dimensions.

An image of the 'zigzag' aperiogon.

Skew polyhedra[edit | edit source]

The five Platonic solids, namely, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron have been known for thousands of years, and that these were the only convex regular polyhedra. Johannes Kepler discovered the small stellated dodecahedron and the great stellated dodecahedron.[1] It was left to Louis Poinsot to discover the great icosahedron and the great dodecahedron.

This was proved to be the complete set by Augustin-Louis Cauchy.

However, to quote Coxeter,

One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedra; infinite, but free from false vertices. When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex [fig], and one consisting of hexagons, four at each vertex [fig]. It was useless to protest that there is no room for more than four squares round a vertex. The trick is, to let the faces go up and down in a kind of zig-zag formation, so that the faces that adjoin a given “horizontal” face lie alternately “above” and “below” it. When I understood this, I pointed out a third possibility: hexagons, six at each vertex [fig].[2]

Skew polyhedra either have skew faces or skew vertex figures. The three infinite regular polyhedra described above (the mucube, muoctahedron, and mutetrahedron) are examples of the latter. The Petrials of these three and the Petrials of the three regular tilings (square tiling, triangular tiling, and hexagonal tiling) are just some of the 27 infinite regular polyhedra with skew faces (those faces being helices or zigzags). And the finite regular polyhedra with skew faces are the Petrials of the nine Platonic and Kepler-Poinsot solids.

Comparison[edit | edit source]

If n is the rank of a polytope or element and m is the dimension of the Euclidean space where it is realized, standard planar polytopes have m = n for every element, skew polytopes have m > n for some element, and tilings have m = n - 1 for the polytope itself.

External links[edit | edit source]

References[edit | edit source]

  1. Mathematical Association of America.https://maa.org/book/export/html/3405556#:~:text=As%20a%20mathematician%2C%20Kepler%20discovered,and%20found%20volumes%20of%20solids
  2. Coxeter, H.S.M. The Beauty of Geometry: Twelve Essays Dover Publications, Inc., 1999, pp. 78-79, italics in original.