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Schläfli symbol

Vertex figureLine segment, 0 < edge length < 2
Related polytopes
DualFlat apeirogon
Abstract & topological properties
Schläfli type{∞}
Dimension vector(0,1)

A zigzag is a regular apeirogon whose vertices are coplanar but not collinear. It may be considered a skew polygon even though it is coplanar.

As a Petrie polygon[edit | edit source]

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[edit | edit source]

Truncated zigzags[edit | edit source]

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

Regular skew polygons[edit | edit source]

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:

  • 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.

However it has several other properties that make it similar to other skew polytopes. As a result it is often classified as skew polytope.

  • 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)
  • It is the result of a non-trivial blend (the flat apeirogon with the dyad).
  • 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.
  • The zigzag appears as a Petrie polygon in the Euclidean tilings. If the zigzag is considered skew, then a polyhedron has either skew faces or skew Petrie polygons. If the zigzag is not considered skew then it must be considered as a special case in the statement of the theorem.
  • The zigzag has the dimension vector . If the zigzag were considered skew, then every non-skew polygon would have a vector. However the only other polygons with a (0,1) dimension vector are the digon and the flat apeirogon neither of which is skew.
  • It is not of full rank.

External Resources[edit | edit source]