Skew apeirogon

Regular skew zig-zag apeirogon
Edges and vertices	∞
Schläfli symbol	{∞}#{ }
Symmetry group	D∞d, [2+,∞], (2*∞)

In geometry, an infinite skew polygon (or skew apeirogon) has vertices that are not all colinear.

Two primary forms have been studied by dimension, 2-dimensional zig-zag skew apeirogons vertices alternating between two parallel lines, and 3-dimensional helical skew apeirogons with vertices on the surface of a cylinder. In 2-dimensions they repeat as glide reflections, as screw axis in 3-dimensions.

Regular skew apeirogon 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 skew zig-zag aperiogon has 2*∞, D_∞dFrieze group symmetry.

The zig-zag regular skew apeirogons exists as Petrie polygons of the three regular tilings of the plane: {4,4}, {6,3}, and {3,6}. These apeirogons have internal angles of 90°, 120°, and 60°, respectively, from the regular polygons within the tilings.

A skew isogonal apeirogon alternates two types of edges with various Frieze group symmetries. Distorted regular skew apeirogons produce zig-zagging isogonal ones with translational symmetry.

Other isogonal skew aperigons have alternate edges parallel to the frieze direction. These all have vertical mirror symmetry in the midpoints of the parallel edges. If both edges are the same length, they can be called quasiregular.

Example quasiregular skew apeirogons can be seen in the Euclidean tilings as truncated Petrie polygons in truncated regular tilings:

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