Zigzag

A zigzag is a regular apeirogon whose vertices are co-planar but not co-linear. It may be considered a skew polygon even though it is co-planar.

As a Petrie polygon
Zigzags appear as Petrie polygons of several polyhedra, in particular in Euclidean tilings, such as the triangular tiling, square tiling, and the hexagonal tiling. As a result they appear as faces of the Petrie duals of these polyhedra: the Petrial triangular tiling, Petrial square tiling, and Petrial hexagonal tiling.

The zigzag also appears as a Petrie polygons in polyhedra made by gyroelongating along an apeirogon. For example a zigzag is one of the Petrie polygons of the apeirogonal antiprism (a gyroelongation of the apeirogonal dihedron). The zigzag then also appears as a face of their Petrie duals.

Truncated zigzags
There are truncated zigzags which are uniform but not regular formed by truncating or quasitruncating the zigzag.

Regular skew polygons
The zigzag may be considered a skew polygon, however it is a marginal example and under some definitions it is not considered skew. The zigzag fails the common criteria for a skew polytope: However it has several other properties that make it similar to other skew polytopes. As a result it is often classified as skew polytope.
 * It resides in a space with dimension equal to its rank. (rank 2 in 2-dimensional Euclidean space)
 * All of its proper elements and figures are non-skew.
 * An abstractly equivalent polytope can fit in a lower dimensional space without losing symmetry. (This is not true of all skew polytopes e.g. mutetrahedron or square duocomb) An equivalent statement is that it is the skew of the flat apeirogon.
 * A regular zigzag cannot have an interior (it partitions the plane, but assigning one side as the interior breaks transitivity). This is a property common to other skew polytopes. In particular the mucube also partitions its space but assigning either side of the partition as the interior of the mucube breaks transitivity. Definitions of polytope that permit the regular zigzag generally also permit other skew polytopes.
 * The zigzag can be viewed as the limit of regular skew $2n$-gons as $n$ goes to infinity, just as the flat apeirogon can be viewed as the limit of regular $n$-gons as $n$ goes to infinity.
 * The zigzag is a special case of the regular helices which are all otherwise skew.