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Usually polygons are considered partially ordered sets, having vertices, edges (sides), and some spanned membran (their area or body). In the early 20^{th} 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 nonflat and still regular "polygons" like finite or infinite zigzags, for infinite helices (with finite flat polygonal projection image), or simply the linear apeirogon.
The bestknown skeletal polygons however are the Petrie polygons. Those are defined to be skeletal polygons contained within regular polytopes as pseudo faces, which outlines in ddimensional subelements a sequence of d+1 consecutive edges and then continues accordingly with the neighbouring ddimensional subelement, where it already had outlined d consecutive edges. As induction start one clearly can take a flat regular polygon. Here the polygon itself and its Petrie polygon coincides. With respect to polyhedra e.g. the Petrie polygons thus are nothing but the equatorial antiprismatic zigzags. In fact, these Petrie polygons quite generally are eqatorial nonflat zigzags, allowing for some projection onto a 2dimensional subspace. Thereby the polytope then will be contained completely within the projection of that zigzag, which by itself then becomes a flat convex regular polygon.
E.g. within regular polytopes xPoQo...oRo = {P,Q,...,R} the length S of the respective Petrie polygon sometimes gets emphasized additionally by an according index, i.e. {P,Q,...,R}_{S}. It should be noted here however that the latter symbol on the other hand is also being reused for according modwraps, i.e. polytopes, which have the same local structure as {P,Q,...,R}, but globally get narrowed down to a fraction S' of the former length S by means of according identifications, thus yielding {P,Q,...,R}_{S'}, cf. e.g. the elliptical polytopes. (Further such modwraps e.g. are considered under the name of regular maps.)
It still took some while until such skeletal polygons where considered to be allowable for polytopal faces too. Rather it is a current topic of interest. The earliest 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 involutoric, i.e. twice its application returns to the original polyhedron again. In fact there you simply have {P,Q}_{R}^{π} = {R,Q}_{P}, which clearly shows this involutoric behaviour.
But much earlier already such skeletal polygons became acceptable for use within vertex figures. This then became the origin for the skew polytopes. Here all 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 allone also other polygonal skeletal structures can be considered too. These are (here given wrt. a polyhedral surface tiling):
etc.
©
Icosahedral graph with emphasized face, Petrie polygon, hole, and 2ndorder Petrie polygon repectively. – Higher ordered ones do not make sense for just 5 faces per vertex.
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,QR} 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,QR} = xPoQoxRo 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, xPoQoxRo happens to be a modwrap of xPoQo, which occurs by identification of each Rth vertex on each hole (in the above defined sense) of xPoQo, i.e. they additionally ask to have xRo for pseudofaces. At least, as far as such a combination of numbers is realizable. Coxeter provides as according requirement (for a representation within 3dimensional Euclidean space): 2 sin(π/P) sin(π/Q) = cos(π/R).
The skewness and the regularity of the vertex figure (antiprismatic zigzag) implies that here Q has to be even generally. As xPoQoxRo and xQoPoxRo 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 3dimensional Euclidean space, but well also within spherical or hyperbolical space of same local dimensionality. In fact an according modwrap x(2P)o(2Q)oxRo 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.
infinite ones (within euclidean space)  

x4o6ox4o, mucube (muc) © as substructure of x2x4o3o4*a = x4o3o4x (chon) x4o3o4x (N → ∞) . . . .  4N  6  3 6  2 6 ++++ x . . . &  2  12N  2 2  1 3 ++++ x4o . . &  4  4  6N *  1 1 x . . x  4  4  * 6N  0 2 ++++ x4o3o . & ♦ 8  12  6 0  N * x4o . x & ♦ 8  12  2 4  * 3N 
x6o4ox4o, muoctahedron (muo) © as substructure of x3x4o2o4*a = o4x3x4o (batch) o4x3x4o (N → ∞) . . . .  6N  4  2 4  4 ++++ . x . . &  2  12N  1 2  3 ++++ o4x . . &  4  4  3N *  2 . x3x .  6  6  * 4N  2 ++++ o4x3x . & ♦ 24  36  6 8  N 
x6o6ox3o, mutetrahedron (mut) © as substructure of x3x3o3o3*a (batatoh) x3x3o3o3*a (N → ∞) . . . .  2N  6  6 6  6 2 ++++ x . . . &  2  6N  2 2  3 1 ++++ x3x . .  6  6  2N *  2 0 x . . o3*a &  3  3  * 4N  1 1 ++++ x3x3o . & ♦ 12  18  4 4  N * x . o3o3*a & ♦ 4  6  0 4  * N 
finite ones (within spherical space)  
x4o6ox3o as substructure of x2x3o3o3*a = x3o3o3x (spid) x3o3o3x . . . .  20  6  6 6  2 6 ++++ x . . . &  2  60  2 2  1 3 ++++ x3o . . &  3  3  40 *  1 1 x . . x  4  4  * 30  0 2 ++++ x3o3o . & ♦ 4  6  4 0  10 * x3o . x & ♦ 6  9  2 3  * 20 
x6o4ox3o as substructure of x3x3o2o3*a = o3x3x3o (deca) o3x3x3o . . . .  30  4  2 4  4 ++++ . x . . &  2  60  1 2  3 ++++ o3x . . &  3  3  20 *  2 . x3x .  6  6  * 20  2 ++++ o3x3x . & ♦ 12  18  4 4  10 
x4o4oxRo = regular map {4,4}_{(R,0)} as substructure of x2xRo2oR*a = xRo xRo ((R,R)dip) xRo xRo . . . .  RR  4  2 4  4 ++++ x . . . &  2  2RR  1 2  3 ++++ xRo . . &  R  R  2R *  2 x . x .  4  4  * RR  2 ++++ xRo x . & ♦ 2R  3R  2 R  2R 
x4o8ox3o as substructure of x2x3o4o3*a = x3o4o3x (spic) x3o4o3x . . . .  144  8  8 8  2 8 ++++ x . . . &  2  576  2 2  1 3 ++++ x3o . . &  3  3  384 *  1 1 x . . x  4  4  * 288  0 2 ++++ x3o4o . & ♦ 6  12  8 0  48 * x3o . x & ♦ 6  9  2 3  * 192 
x8o4ox3o as substructure of x4x3o2o3*a = o3x4x3o (cont) o3x4x3o . . . .  288  4  2 4  4 ++++ . x . . &  2  576  1 2  3 ++++ o3x . . &  3  3  192 *  2 . x4x .  8  8  * 144  2 ++++ o3x4x . & ♦ 24  36  8 6  48 
x8o3ox3o as substructure of o3x3o4o (ico) o3x3o4o . . . .  24 ♦ 8  4 8  4 2 ++++ . x . .  2  96  1 2  2 1 ++++ o3x . .  3  3  32 *  2 0 . x3o .  3  3  * 64  1 1 ++++ o3x3o . ♦ 6  12  4 4  16 * . x3o4o ♦ 6  12  0 8  * 8 
x4o3ox4o as substructure of x2x4o3o = x o3o4x (tes) x o3o4x . . . .  16 ♦ 1 3  3 3  3 1 ++++ x . . .  2  8 *  3 0  3 0 . . . x  2  * 24  1 2  2 1 ++++ x . . x  4  2 2  12 *  2 0 . . o4x  4  0 4  * 12  1 1 ++++ x . o4x ♦ 8  4 8  4 2  6 * . o3o4x ♦ 8  0 12  0 6  * 2(however this one decomposes into 2 unconnected cubes) 
x3o4ox4o as substructure of x2x3o4o = x x3o4o (ope) x x3o4o . . . .  12 ♦ 1 4  4 4  4 1 ++++ x . . .  2  6 *  4 0  4 0 . x . .  2  * 24  1 2  2 1 ++++ x x . .  4  2 2  12 *  2 0 . x3o .  3  0 3  * 16  1 1 ++++ x x3o . ♦ 6  3 6  3 2  8 * . x3o4o ♦ 6  0 12  0 8  * 2(however this one decomposes into 2 unconnected octs) 
x3o6ox3o as substructure of o3x3o3o (rap) o3x3o3o . . . .  10 ♦ 6  3 6  3 2 ++++ . x . .  2  30  1 2  2 1 ++++ o3x . .  3  3  10 *  2 0 . x3o .  3  3  * 20  1 1 ++++ o3x3o . ♦ 6  12  4 4  5 * . x3o3o ♦ 4  6  0 4  * 5(however this one results in 5 tipconnected tets) 
some infinite ones (within hyperbolical space)  
x6o6ox4o as substructure of x3x4o3o4*a (x3x4o3o4*a) x3x4o3o4*a (N → ∞) . . . .  4N  6  6 6  6 2 ++++ x . . . &  2  12N  2 2  3 1 ++++ x3x . .  6  6  4N *  2 0 x . . o4*a &  4  4  * 6N  1 1 ++++ x3x4o . & ♦ 24  36  8 6  N * x . o3o4*a & ♦ 8  12  0 6  * N 
x8o8ox3o as substructure of o3x4x3o4*a (o3x4x3o4*a) o3x4x3o4*a (N → ∞) . . . .  3N  8  8 8  8 2 ++++ . x . . &  2  12N  2 2  3 1 ++++ o3x . . &  3  3  8N *  1 1 . x4x .  8  8  * 3N  2 0 ++++ o3x4x . & ♦ 24  36  8 6  N * o3x . o4*a & ♦ 6  12  8 0  * N 
... 
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 2Rth 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 2Rgonal faces.
Other infinite polyhedra with flat faces and a single skew vertex figure type only, even when restricted to 3dimensional 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.
just some examples ...
Just as infinite regular skew polyhedra (those muhedra) where obtained from euclidean honeycombs it is straight forward to obtain infinte regular skew polychora (i.e. muchora) as substructures of euclidean tetracombs as outlined by Klitzing in 2020. The following first 3 had be derived by the author in those days, the next 2 have been found a year later by Mecejide.
Similar to the 3D case, where {P,QR} = xPoQoxRo described structures with Pgonal faces meeting Q to a vertex with Rgonal pseudofaces, those symbols too could be extrapolated as well. We thus define {P,Q,RS,T} = xPoQoRoxSoTo to have {P,Q} = xPoQo cells meeting R to an edge and with {S,T} = xSoTo pseudocells. From the connectivity however it follows that P and S here are bound to have the same value always.
just some infinite ones (within euclidean space)  

• 16 tets per vertex :
x3o3o4oo3x4o as substructure of o4x3o3o4o (rittit) o4x3o3o4o (N → ∞) . . . . .  4N  12  6 24  12 16  8 2 +++++ . x . . .  2  24N  1 4  4 4  4 1 +++++ o4x . . .  4  4  6N *  4 0  4 0 . x3o . .  3  3  * 32N  1 2  2 1 +++++ o4x3o . . ♦ 12  24  6 8  4N *  2 0 . x3o3o . ♦ 4  6  0 4  * 16N  1 1 +++++ o4x3o3o . ♦ 32  96  24 64  8 16  N * . x3o3o4o ♦ 8  24  0 32  0 16  * Nthe verf of this substructure happens to be x3o4ox4o. 
• 12 octs per vertex :
x3o4o3ox3o3o as substructure of o3o4o3x3o (icot) o3o4o3x3o (N → ∞) . . . . .  12N ♦ 16  24 8  12 12  2 6 +++++ . . . x .  2  96N ♦ 3 1  3 3  1 3 +++++ . . o3x .  3  3  96N *  2 1  1 2 . . . x3o  3  3  * 32N  0 3  0 3 +++++ . o4o3x . ♦ 6  12  8 0  24N *  1 1 . . o3x3o ♦ 6  12  4 4  * 24N  0 2 +++++ o3o4o3x . ♦ 24  96  96 0  24 0  N * . o4o3x3o ♦ 24  96  64 32  8 16  * 3Nthe verf of this substructure happens to be x4o3ox4o. 
• 64 tets per vertex :
x3o3o8ox3o3o as substructure of x3o3o *b3o4o (hext) x3o3o *b3o4o (N → ∞) . . . . .  N ♦ 24  96  32 64  16 8 +++++ x . . . .  2  12N ♦ 8  4 8  4 2 +++++ x3o . . .  3  3  32N  1 2  2 1 +++++ x3o3o . . ♦ 4  6  4  8N *  2 0 x3o . *b3o . ♦ 4  6  4  * 16N  1 1 +++++ x3o3o *b3o . ♦ 8  24  32  8 8  2N * x3o . *b3o4o ♦ 8  24  32  0 16  * Nthe verf of this substructure happens to be x8o3ox3o. 
• 30 octs per vertex :
x3o4o6ox3o3o as substructure of x3o3o3o3o3*a (cypit) x3o3o3o3o3*a (N → ∞) . . . . .  N ♦ 20  60  40 30  10 20 +++++ x . . . .  2  10N ♦ 6  6 6  2 6 +++++ x3o . . . &  3  3  20N  2 2  1 3 +++++ x3o3o . . & ♦ 4  6  4  10N *  1 1 x3o . . o3*a ♦ 6  12  8  * 5N  0 2 +++++ x3o3o3o . & ♦ 5  10  10  5 0  2N * x3o3o . o3*a & ♦ 10  30  30  5 5  * 2Nthe verf of this substructure happens to be x4o6ox3o. 
• 16 octs per vertex :
x3o4o4ox3o4o as substructure of o4o3x3o4o (icot) o4o3x3o4o (N → ∞) . . . . .  3N ♦ 16  32  8 16  8 +++++ . . x . .  2  24N ♦ 4  2 4  4 +++++ . o3x . . &  3  3  32N  1 2  3 +++++ o4o3x . . & ♦ 6  12  8  4N *  2 . o3x3o . ♦ 6  12  8  * 8N  2 +++++ o4o3x3o . & ♦ 24  96  96  8 16  Nthe verf of this substructure happens to be x4o4ox4o. 

finite ones (within spherical space)  
• 20 tets per vertex :
x3o3o6ox3o3o as substructure of x3o3o *b3o3o (hin) x3o3o *b3o3o . . . . .  16 ♦ 10  30  10 20  5 5 +++++ x . . . .  2  80 ♦ 6  3 6  3 2 +++++ x3o . . .  3  3  160  1 2  2 1 +++++ x3o3o . . ♦ 4  6  4  40 *  2 0 x3o . *b3o . ♦ 4  6  4  * 80  1 1 +++++ x3o3o *b3o . ♦ 8  24  32  8 8  10 * x3o . *b3o3o ♦ 5  10  10  0 5  * 16the verf of this substructure happens to be x3o6ox3o. 
Miss Krieger outlined shortly after those 2020 finds this finite muteron.
just some finite ones (within spherical space) 

• 80 pens per vertex :
x3o3o3o6ox3o3o3o as substructure of x3o3o3o3o *c3o (jak) x3o3o3o3o *c3o . . . . . .  27 ♦ 16  80  160  80 40  16 10 ++++++ x . . . . .  2  216 ♦ 10  30  20 10  5 5 ++++++ x3o . . . .  3  3  720 ♦ 6  6 3  2 3 ++++++ x3o3o . . . ♦ 4  6  4  1080  2 1  1 2 ++++++ x3o3o3o . . ♦ 5  10  10  5  432 *  1 1 x3o3o . . *c3o ♦ 5  10  10  5  * 216  0 2 ++++++ x3o3o3o3o . ♦ 6  15  20  15  6 0  72 * x3o3o3o . *c3o ♦ 10  40  80  80  16 16  * 27the verf of this substructure happens to be x3o3o6ox3o3o. 
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