Skew polytope

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A skew polytope is a polytope of rank n whose vertices do not all lie in an n-dimensional subspace. More generally, a polytope can be called skew if any of its elements satisfies this description. A non-skew polytope is called planar.

Skew polygons

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

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.

Petrie polygons

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

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

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 great icosahedron and the great stellated dodecahedron. There were two other polyhedra that Kepler overlooked (as they don’t obey the Euler equality, ), and it was left to Louis Poinsot to discover the small stellated dodecahedron 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[1]. 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]
  1. Places where, due to the self-intersecting nature of some of the non-convex polytopes, apparent vertices can be found, but which do not demark the vertices of the constituent facets.
  2. Coxeter, H.S.M. The Beauty of Geometry: Twelve Essays Dover Publications, Inc., 1999, pp. 78-79, italics in original.