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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 polygons (apeirogons) 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 (and regular skew apeirogons in two dimensions) are always zig-zag.
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.
A regular skew n-gon can be given a Schläfli 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.
Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes. For example, the five Platonic solids have 4-, 6-, and 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around their respective projective envelopes. The tetrahedron and the octahedron include all the vertices in their respective zig-zag skew polygons, and can be seen as a digonal antiprism and a triangular antiprism respectively.
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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 polygons (apeirogons) 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 (and regular skew apeirogons in two dimensions) are always zig-zag.
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.
A regular skew n-gon can be given a Schläfli 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.
Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes. For example, the five Platonic solids have 4-, 6-, and 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around their respective projective envelopes. The tetrahedron and the octahedron include all the vertices in their respective zig-zag skew polygons, and can be seen as a digonal antiprism and a triangular antiprism respectively.