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Skew polyhedra

Usually polygons are considered partially ordered sets, having vertices, edges (sides), and some spanned membran (their area or body). In the early 20th century Coxeter and Petrie came up with the concept of skeletal polygons. There only the vertices and edges remain to be considered, whereas the existance of a spanned (flat) membran is no longer required. This then also allows for non-flat and still regular "polygons" like finite or infinite zig-zags, for infinite helices (with finite flat polygonal projection image), or simply the linear apeirogon.

The best-known skeletal polygons however are the Petrie polygons. Those are defined to be skeletal polygons contained within regular polytopes as pseudo faces, which outlines in d-dimensional subelements a sequence of d+1 consecutive edges and then continues accordingly with the neighbouring d-dimensional subelement, where it already had outlined d consecutive edges. With respect to polyhedra the Petrie polygons e.g. are nothing but the equatorial antiprismatic zig-zags. In fact, these Petrie polygons generally are eqatorial non-flat zig-zags, allowing for some projection onto a 2-dimensional subspace, where the polytope then will be contained completely within the projection of tha zig-zag, which by itself then becomes a flat convex regular polygon.

It still took some while until such skeletal polygons where considered to be allowable for polytopal faces. Rather it is a current topic of interest. The earlies according acception maybe was within the definition of the Petrie dual polyhedra. Those are defined to use the same edge skeleton then the starting regular polyhedron, but replaces the former faces by the respective Petrie polygons. This operation then happens to be infact involutoric, i.e. twice its application returns to the original polyhedron again.

But much earlier already such skeletal polygons became acceptable for use within vertex figures. This then became the origin for the skew polytopes. Here the face polygons still remain flat. One further restricts again for local finiteness, esp. dense vertex figures would not be allowed and therefore infinit skeletal polygons are excluded usually as vertex figures.

Besides the Petrie polygons also other polygonal structures are considered. These are (here given wrt. a polyhedral surface tiling):

face
closed or unbounded sequence of edges (sides), making at every vertex the sharpest possible bend, being done into the same direction always
Petrie polygon
closed or unbounded sequence of edges (sides), making at every vertex the sharpest possible bend, being done alternatingly in both directions
hole
closed or unbounded sequence of edges (sides), making at every vertex the 2nd sharpest possible bend, being done into the same direction always
2nd order Petrie polygon
closed or unbounded sequence of edges (sides), making at every vertex the 2nd sharpest possible bend, being done alternatingly in both directions
3rd order hole
closed or unbounded sequence of edges (sides), making at every vertex the 3rd sharpest possible bend, being done into the same direction always

etc.

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Icosahedral graph with emphasized face, Petrie polygon, hole, and 2nd-order Petrie polygon repectively. – Higher ordered ones do not make sense for just 5 faces per vertex.



Regular skew Petrie-Coxeter polyhedra

Petrie and Coxeter quite early found 3 regular skew polyhedra, all having flat regular convex faces and finite regular skew polygonal vertex figures. Those all happen to be apeirohedra (infinite count of faces) and show up holes (within common sense), providing higher genus patterns. Coxeter writes {P,Q|R} for such a structure. Here P is the (flat) polygonal type of the faces, Q the (skew) polygonal type of the vertex figures, and R the (flat) polygonal type of the tunnel cross section (hole).

Obviously {P,Q|R} = xPoQo|R and (flat) {P,Q} = xPoQo have the same incidence structure. But the former has some further symmetry applied, which circles around the tunnels (holes). In fact, xPoQo|R happens to be a modwrap of xPoQo, which occurs by identification of each R-th vertex on each hole (in the above defined sense) of xPoQo. At least, as far as such a combination of numbers is realizable. Coxeter provides as according requirement (for a representation within 3-dimensional Euclidean space) 2 sin(π/P) sin(π/Q) = cos(π/R).

x4o6o|4     x6o4o|4     x6o6o|3 ©

The skewness and the regularity of the vertex figure (antiprismatic zigzag) implies that here Q has to be even generally. As xPoQo|R and xQoPo|R generally come in dual pairs, the same holds true for P.

Generally it happens that such skew representations of infinite regular polyhedra with flat regular polygons not only occur as embeddings within 3-dimensional Euclidean space, but well also within spherical or hyperbolical space of same local dimensionality. In fact an according modwrap x(2P)o(2Q)o|R generally occurs as a facial substructure of xPxRoQoR*a. And the above mentioned appearance in dual pairings then reflects itself just in the interchange of the node symbol types.

The easiest spherical space examples e.g. here obviously are all these {4,4|R}, which then just describe the torical squares, i.e. lacing squares of the R-prismatic cells of the general uniform (R,R)-duoprism.

Wythoffian skew polyhedra with flat faces

A direct application of Wythoff's construction advice might not be too obvious here. But the according local operations in the sense of Stott expansions surely apply. Because of the existance of skew vertex figures however, this generally would produce not infinite polyhedra with flat faces only. But a single generally applicable exception each could be listed here, none the less. It is the modwrap of x(2P)o(2Q)x, when identifying each 2R-th vertex on each hole. That one then likewise occurs as according facial substructure of xPxRxQxR*a. In fact there it omits, beside all cell bodies for sure, all 2R-gonal faces.

x4o6x|8 ©     x6o6x|6

Uniform skew polyhedra with flat faces

Other infinite polyhedra with flat faces and a single skew vertex figure type only, even when restricted to 3-dimensional eucidean embedding space, so far are still not being classified. Many of them simply are obtained as mirror periodic packings of one ore two uniform polyhedra.

using a single uniform solid as building unit

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using two uniform solids as building unit

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not related to a decomposition into convex uniform solids

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