Regular skew polygon

In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The interior surface (or area) of such a polygon is not uniquely defined.

Skew infinite polygons (apeirogons) have vertices which are not all collinear.

A zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, and thus must be even-sided.

Regular skew polygon in 3 dimensions (and regular skew apeirogons in 2 dimensions) are always zig-zag.

A regular skew polygon is isogonal with equal edge lengths. In 3 dimensions a regular skew polygon is a zig-zag skew (or antiprismatic polygon), with vertices alternating between two parallel planes. The sides of an n-antiprism can define a regular skew 2n-gons.

A regular skew n-gonal can be given a symbol {p}#{ } as a blend of a regular polygon, {p} and an orthogonal line segment, { }. The symmetry operation between sequential vertices is glide reflection.

Examples are shown on the uniform square and pentagon antiprisms. The star antiprisms also generate regular skew polygons with different connection order of the top and bottom polygons. The filled top and bottom polygons are drawn for structural clarity, and are not part of the skew polygons.

A regular compound skew 2n-gon can be similarly constructed by adding a second skew polygon by a rotation. These shares the same vertices as the prismatic compound of antiprisms.

Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes. For example, the 5 Platonic solids have 4, 6, and 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around the projective envelope. The tetrahedron and octahedron include all the vertices in the zig-zag skew polygon and can be seen as a digonal and a triangular antiprisms respectively.

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