 | This articlepossibly containsoriginal research. Pleaseimprove it byverifying the claims made and addinginline citations. Statements consisting only of original research should be removed.(December 2019) (Learn how and when to remove this message) |
Ingeometry, aninfiniteskew polygon orskewapeirogon is an infinite 2-polytope with vertices that are not allcolinear.Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines.Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of acylinder.
Regular infinite skew polygons exist in thePetrie polygons of the affine and hyperbolicCoxeter groups. They are constructed a single operator as the composite of all the reflections of the Coxeter group.
| Regular zig-zag skew apeirogon | |
|---|---|
 | |
| Edges andvertices | ∞ |
| Schläfli symbol | {∞}#{ } |
| Symmetry group | D∞d, [2+,∞], (2*∞) |
A regular zig-zag skew apeirogon has (2*∞), D∞dFrieze group symmetry.
Regular zig-zag skew apeirogons exist asPetrie polygons of the three regular tilings of the plane: {4,4}, {6,3}, and {3,6}. These regular zig-zag skew apeirogons haveinternal angles of 90°, 120°, and 60° respectively, from the regular polygons within the tilings:
 |
Anisotoxal apeirogon has one edge type, between two alternating vertex types. There's a degree of freedom in theinternal angle, α. {∞α} is thedual polygon of an isogonal skew apeirogon.
| {∞0°} |  |
| {∞30°} |  |
Anisogonal skew apeirogon alternates two types of edges with variousFrieze group symmetries. Distorted regular zig-zag skew apeirogons produce isogonal zig-zag skew apeirogons with translational symmetry:
| p1, [∞]+, (∞∞), C∞ | |
|---|---|
  |   |
Other isogonal skew apeirogons have alternate edges parallel to the Frieze direction. These isogonal elongated skew apeirogons have vertical mirror symmetry in the midpoints of the edges parallel to the Frieze direction:
| p2mg, [2+,∞], (2*∞), D∞d | ||
|---|---|---|
  |   |   |
An isogonal elongated skew apeirogon has two different edge types; if both of its edge types have the same length: it can't be called regular because its two edge types are still different ("trans-edge" and "cis-edge"), but it can be called quasiregular.
Example quasiregular elongated skew apeirogons can be seen as truncated Petrie polygons in truncated regular tilings of the Euclidean plane:
Infinite regular skew polygons are similarly found in the Euclidean plane and in thehyperbolic plane.
Hyperbolic infinite regular skew polygons also exist asPetrie polygons zig-zagging edge paths on allregular tilings of the hyperbolic plane. And again like in the Euclidean plane, hyperbolic infinite quasiregular skew polygons can be constructed as truncated Petrie polygons within the edges of all truncated regular tilings of the hyperbolic plane.
| {3,7} | t{3,7} |
|---|---|
 Regular skew |  Quasiregular skew |
{∞} # {3} An infinite regularhelical polygon (drawn inperspective) |
An infinitehelical (skew) polygon can exist in three dimensions, where the vertices can be seen as limited to the surface of acylinder. The sketch on the right is a 3D perspective view of such an infinite regular helical polygon.
This infinite helical polygon can be mostly seen as constructed from the vertices in an infinite stack ofuniformn-gonalprisms orantiprisms, although in general the twist angle is not limited to an integer divisor of 180°. An infinite helical (skew) polygon hasscrew axis symmetry.
An infinite stack ofprisms, for example cubes, contain an infinite helical polygon across the diagonals of the square faces, with a twist angle of 90° and with a Schläfli symbol {∞} # {4}.
An infinite stack of antiprisms, for exampleoctahedra, makes infinite helical polygons, 3 here highlighted in red, green, and blue, each with a twist angle of 60° and with a Schläfli symbol {∞} # {6}.
A sequence of edges of aBoerdijk–Coxeter helix can represent infinite regular helical polygons with an irrational twist angle:
A stack of rightprisms can generate isogonal helical apeirogons alternating edges around axis, and along axis; for example a stack of cubes can generate this isogonal helical apeirogon alternating red and blue edges:
Similarly an alternating stack of prisms and antiprisms can produce an infinite isogonal helical polygon; for example, a triangular stack of prisms and antiprisms with an infinite isogonal helical polygon:
An infinite isogonal helical polygon with an irrational twist angle can also be constructed fromtruncated tetrahedra stacked like aBoerdijk–Coxeter helix, alternating two types of edges, between pairs of hexagonal faces and pairs of triangular faces:
