Infinite skew polygon

In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. 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 a cylinder. Regular infinite skew polygons exist in the Petrie polygons of the affine and hyperbolic Coxeter groups. They are constructed a single operator as the composite of all the reflections of the Coxeter group. A regular zig-zag skew apeirogon has (2*∞), D∞d Frieze group symmetry. Regular zig-zag skew apeirogons exist as Petrie polygons of the three regular tilings of the plane: {4,4}, {6,3}, and {3,6}. These regular zig-zag skew apeirogons have internal angles of 90°, 120°, and 60° respectively, from the regular polygons within the tilings: An isotoxal apeirogon has one edge type, between two alternating vertex types. There's a degree of freedom in the internal angle, α. {∞α} is the dual polygon of an isogonal skew apeirogon. An isogonal skew apeirogon alternates two types of edges with various Frieze group symmetries. Distorted regular zig-zag skew apeirogons produce isogonal zig-zag skew apeirogons with translational symmetry: 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: 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 the hyperbolic plane.