Zigzag

Zigzag
Rank	2
Type	Regular
Space	Euclidean
Notation
Schläfli symbol	$\{\infty\}\#\{\}$
Elements
Edges	$N$
Vertices	$N$
Vertex figure	Line segment, 0 < edge length < 2
Related polytopes
Dual	Zigzag
Topological properties
Orientable	Yes
Properties
Convex	No

This article is about the infinite polygon. For the path on the vertices of a polytope, see Zigzag walk.

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

Three regular tilings with a zigzag highlighed in red.

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.

Related polygons

Truncated zigzags

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

Regular skew polygons

Just as the flat apeirogon can be viewed as the limit of regular n-gons as n goes to infinity, the zigzag can be viewed as the limit of regular skew n-gons as n goes to infinity. This results in the zigzag often being classified as a skew polygon despite it being planar.

External Resources

• Wikipedia Contributors. "Infinite skew polygon".