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Ingeometry, askew polygon is a closedpolygonal chain inEuclidean space. It is afigure similar to apolygon except itsvertices are not allcoplanar.[1] While a polygon is ordinarily defined as aplane figure, theedges and vertices of a skew polygon form aspace curve. Skew polygons must have at least four vertices. Theinteriorsurface and corresponding area measure of such a polygon is not uniquely defined.
Skew infinite polygons (apeirogons) have vertices which are not all colinear.
Azig-zag skew polygon orantiprismatic polygon[2] 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.
Aregular 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 skewn-gon can be given aSchläfli symbol{p}#{} as ablend of aregular polygonp and an orthogonalline segment { }.[3] The symmetry operation between sequential vertices isglide reflection.
Examples are shown on the uniform square and pentagon antiprisms. Thestar 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.
| Skew square | Skew hexagon | Skew octagon | Skew decagon | Skew dodecagon | ||
| {4}#{ } | {6}#{ } | {8}#{ } | {10}#{ } | {5}#{ } | {5/2}#{ } | {12}#{ } |
|  |  |  |  |  |  |
| s{2,4} | s{2,6} | s{2,8} | s{2,10} | sr{2,5/2} | s{2,10/3} | s{2,12} |
Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes. For example, the fivePlatonic solids have 4-, 6-, and 10-sided regular skew polygons, as seen in theseorthogonal projections with red edges around their respectiveprojective 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.
Aregular skew polyhedron has regular polygon faces, and a regular skew polygonvertex figure.
Three infinite regular skew polyhedra arespace-filling in 3-space; othersexist in 4-space, some within theuniform 4-polytopes.
| {4,6|4} | {6,4|4} | {6,6|3} |
|---|---|---|
 Regular skew hexagon {3}#{ } |  Regular skew square {2}#{ } |  Regular skew hexagon {3}#{ } |
In 4 dimensions, a regular skew polygon can have vertices on aClifford torus and related by aClifford displacement. Unlike zig-zag skew polygons, skew polygons on double rotations can include an odd-number of sides.
ThePetrie polygons of theregular 4-polytopes define regular zig-zag skew polygons. TheCoxeter number for eachcoxeter group symmetry expresses how many sides a Petrie polygon has. This is 5 sides for a5-cell, 8 sides for atesseract and16-cell, 12 sides for a24-cell, and 30 sides for a120-cell and600-cell.
When orthogonally projected onto theCoxeter plane, these regular skew polygons appear as regular polygon envelopes in the plane.
| A4, [3,3,3] | B4, [4,3,3] | F4, [3,4,3] | H4, [5,3,3] | ||
|---|---|---|---|---|---|
| Pentagon | Octagon | Dodecagon | Triacontagon | ||
 5-cell {3,3,3} |  tesseract {4,3,3} |  16-cell {3,3,4} |  24-cell {3,4,3} |  120-cell {5,3,3} |  600-cell {3,3,5} |
Then-nduoprisms and dualduopyramids also have 2n-gonal Petrie polygons. (Thetesseract is a 4-4 duoprism, and the16-cell is a 4-4 duopyramid.)
| Hexagon | Decagon | Dodecagon | |||
|---|---|---|---|---|---|
 3-3 duoprism |  3-3 duopyramid |  5-5 duoprism |  5-5 duopyramid |  6-6 duoprism |  6-6 duopyramid |