Skew polygon

In geometry, a skew polygon is a closed polygonal chain in Euclidean space. It is a figure similar to a polygon except its vertices are not all coplanar. While a polygon is ordinarily defined as a plane figure, the edges and vertices of a skew polygon form a space curve. Skew polygons must have at least four vertices. The interior surface and corresponding area measure of such a polygon is not uniquely defined.
Skew [infinite polygon]s have vertices which are not all colinear.
A zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, and thus must be even-sided.
Regular skew polygons in 3 dimensions are always zig-zag.

Skew polygons in three dimensions

A regular skew polygon is a faithful symmetric realization of a polygon in dimension greater than 2. In 3 dimensions a regular skew polygon has vertices alternating between two parallel planes.
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Regular skew polygon as vertex figure of regular skew polyhedron

A regular skew polyhedron has regular polygon faces, and a regular skew polygon vertex figure.
Three infinite regular skew polyhedra are space-filling in 3-space; others regular skew polyhedron#Finite regular [skew polyhedra of 4-space|exist in 4-space], some within the uniform 4-polytopes.


Regular skew hexagon	Regular skew square	Regular skew hexagon

Regular skew polygons in four dimensions

In 4 dimensions, a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike zig-zag skew polygons, skew polygons on double rotations can include an odd-number of sides.
The Petrie polygons of the regular 4-polytopes define regular zig-zag skew polygons. The Coxeter number for each coxeter group symmetry expresses how many sides a Petrie polygon has. This is 5 sides for a 5-cell, 8 sides for a tesseract and 16-cell, 12 sides for a 24-cell, and 30 sides for a 120-cell and 600-cell.
When orthogonally projected onto the Coxeter plane, these regular skew polygons appear as regular polygon envelopes in the plane.
The n-''n duoprisms and dual duopyramids also have 2n''-gonal Petrie polygons.