Skew polygon

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The (red) side edges of tetragonal disphenoid represent a regular zig-zag skew quadrillateral.

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[1] has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygon existing in 3-dimensions (and regular skew apeirogons in 2-dimensions) are always zig-zag.

Zig-zag skew polygon in 3-dimensions[edit]

A uniform n-gonal antiprism has a 2n-sided regular skew polygon defined along its side edges.

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-polygons.

A regular skew n-gonal can be given a symbol {p}#{ } as a blend of a regular polygon, {p} and an orthogonal line segment, { }.[2] 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.

Regular skew zig-zag decagons
{4}#{ } {5}#{ } {5/2}#{ } {5/3}#{ }
Skew polygon in square antiprism.png
sr{2,4}
Regular skew polygon in pentagonal antiprism.png
sr{2,5}
Regular skew polygon in pentagrammic antiprism.png
sr{2,5/2}
Regular skew polygon in pentagrammic crossed-antiprism.png
sr{2,5/3}

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.

Petrie polygons.png

The regular skew polyhedron have regular faces, and regular skew polygon vertex figures. Three are infinite 3-space filling and others exist in 4-space, some within the uniform 4-polytope.

Skew Vertex figures of the 3 infinite regular skew polyhedra
{4,6|4} {6,4|4} {6,6|3}
Six-square skew polyhedron-vf.png
Regular skew hexagon
{3}#{ }
Four-hexagon skew polyhedron-vf.png
Regular skew square
{2}#{ }
Six-hexagon skew polyhedron-vf.png
Regular skew hexagon
{3}#{ }

Isogonal zig-zag skew polygons[edit]

An isogonal skew polygon is a skew polygon with one type of vertex, connected by two types of edges. Isogonal skew polygons with equal edge lengths can also be considered quasiregular. It is similar to a zig-zag skew polygon, existing on two planes, except allowing one edge to cross to the opposite plane, and the other edge to stay on the same plane.

Isogonal skew polygons can be defined on even-sided n-gonal prisms, alternatingly following an edge of one side polygon, and moving between polygons. For example on the vertices of a cube. Vertices alternate between top and bottom squares with red edges between sides, and blue edges along each side.

Isogonal skew octagon on cube2.png
Cube, square-diagonal
Isogonal skew octagon on cube.png
Cube
Isogonal skew octagon on crossed-cube.png
Crossed cube
Isogonal skew octagon on hexagonal prism.png
Hexagonal prism
Isogonal skew octagon on hexagonal prism2.png
Hexagonal prism

Regular skew polygons in 4-dimensions[edit]

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-polytope define regular 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.

A4, [3,3,3] B4, [4,3,3] F4, [3,4,3] H4, [5,3,3]
Pentagon Octagon Dodecagon Triacontagon
4-simplex t0.svg
5-cell
{3,3,3}
4-cube graph.svg
tesseract
{4,3,3}
4-orthoplex.svg
16-cell
{3,3,4}
24-cell t0 F4.svg
24-cell
{3,4,3}
120-cell graph H4.svg
120-cell
{5,3,3}
600-cell graph H4.svg
600-cell
{3,3,5}

The n-n duoprism and dual duopyramids also have 2n-gonal petrie polygons. (The tesseract is a 4-4 duoprism, and the 16-cell is a 4-4 duopyramid.)

Hexagon Decagon Dodecagon
3-3 duoprism ortho-Dih3.png
3-3 duoprism
3-3 duopyramid ortho.png
3-3 duopyramid
5-5 duoprism ortho-Dih5.png
5-5 duoprism
5-5 duopyramid ortho.png
5-5 duopyramid
6-6 duoprism ortho-3.png
6-6 duoprism
6-6 duopyramid ortho-3.png
6-6 duopyramid

Skew apeirogon[edit]

An infinite skew polygon, also called a skew apeirogon has vertices that are not all colinear.

Two primary forms have been studied by dimension, 2-dimensional zig-zag skew apeirogons vertices alternating between two parallel lines, and 3-dimensional helical skew apeirogons with vertices on the surface of a cylinder. In 2-dimensions they repeat as glide reflections,[3] as screw axis in 3-dimensions, and rotary translation in general.

Regular skew apeirogon exist in the petrie polygons of the affine and hyperbolic Coxeter groups.

Regular skew apeirogons in 2-dimensions[edit]

Regular skew zig-zag apeirogon
Regular apeirogon zig-zag.png
Edges and vertices
Schläfli symbol {∞}#{ }
Symmetry group D∞d, [2+,∞], (2*∞)
The angled edges of an apeirogonal antiprism represent a regular zig-zag aperiogon.

A regular skew zig-zag aperiogon has 2*∞, D∞d Frieze group symmetry.

The zig-zag regular skew apeirogon exists as Petrie polygons of the 3 regular tilings of the plane: {4,4}, {6,3}, and {3,6}. These apeirogons have a internal angles of 90°, 120°, and 60° respectively, from the regular polygons within the tilings.

Petrie polygons of regular tilings of the plane
Petrie polygons of regular tilings.png

Isogonal skew apeirogons[edit]

A skew isogonal apeirogon alternates two types of edges with various Frieze group symmetries. Distorted regular skew apeirogons produce zig-zagging isogonal ones with translational symmetry.

isogonal skew apeirogons in 2-dimensions (zig-zag)
p1, [∞]+, (∞∞), C
Isogonal apeirogon skew-equal.png
Isogonal apeirogon skew-unequal.png
Isogonal apeirogon.png
Isogonal apeirogon skew-unequal-backwards.png

Other isogonal skew aperigons have alternate edges parallel to the frieze direction. These all have vertical mirror symmetry in the midpoints of the parallel edges. If both edges are the same length, they can be called quasiregular.

Isogonal skew apeirogons in 2-dimensions (elongated)
p2mm, [2,∞], (*∞22), D∞h p2mg, [2+,∞], (2*∞), D∞d
Isogonal apeirogon2.png
Isogonal apeirogon2-rectangle.png
Isogonal apeirogon2a.png
Isogonal apeirogon2b.png
Isogonal apeirogon2c.png
Isogonal apeirogon2d.png

Example quasiregular skew apeirogons can be seen in the Euclidean tilings as truncated Petrie polygons in truncated regular tilings:

Quasiregular skew apeirogon in truncated tilings.png

Hyperbolic skew apeirogons[edit]

In hyperbolic geometry, regular skew apeirogons are similarly found like in the Euclidean plane.

Hyperbolic regular skew apeirogons also exist as Petrie polygon zig-zagging edge paths on all of the regular tilings of the hyperbolic plane. And again like Euclidean space, hyperbolic quasiregular skew apeirogons can be constructed as truncated petrie polygons within the edges of a truncated regular tiling.

Regular and uniform tilings with apeirogons
{3,7} t{3,7}
Order-7 triangular tiling petrie polygon.png
Regular skew
Quasiregular skew apeirogon in truncated order-7 triangular tiling.png
Quasiregular skew

Helical apeirogons in 3-dimensions[edit]

A regular apeirogon in 3-dimensions
Triangular helix.png
A regular helical skew polygon
(drawn in perspective)

A helical skew apeirogon can exist in three dimensions, where the vertices can be seen as limited to the surface of a cylinder. The sketch on the right is a 3D perspective view of such a regular helical apeirogon.

This apeirogon can be most seen as constructed from the vertices in an infinite stack of uniform n-gonal uniform prisms or antiprisms, although in general the twist angle is not limited to an integer divisor of 180°. A helical skew apeirogon has screw axis symmetry.

An infinite stack of prisms, for example cubes, contain a helical apeirogon across the diagonals of the square faces, with a twist angle of 90°.

Cube stack diagonal-face helix apeirogon.png

An infinite stack of antiprisms, for example octahedra, makes helical apeirogons, 3 here highlighted in red, green and blue, each with a twist angle of 60°.

Octahedron stack helix apeirogons.png

A sequence of edges of a Boerdijk–Coxeter helix can represent regular helical apeirogons with an irrational twist angle:

Coxeter helix edges.png

Isogonal helical apeirogons[edit]

A stack of right prisms can generate isogonal helical aperigon alternating edges around axis, and along axis, here for example with square prisms, alternating red and blue edges:

Cubic stack isogonal helical apeirogon.png

Similarly an alternating stack of prisms and antiprisms can produce an isogonal helical apeirogon, here for example, a triangular one with one shown:

Elongated octahedron stack isogonal helical apeirogon.png

An isogonal helical apeirogon with an irrational twist angle can also be constructed from truncated tetrahedra stacked like a Boerdijk–Coxeter helix, alternating two types of edges, between pairs of hexagonal and pairs of triangular faces:

Quasiregular helix apeirogon in truncated Coxeter helix.png

See also[edit]

References[edit]

  1. ^ Regular complex polytopes , p. 6
  2. ^ Abstract Regular Polytopes, p.217
  3. ^ Coxeter, H. S. M. and Moser, W. O. J. (1980), p.54 5.2 The Petrie polygon
  • McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0  p.25
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  "Skew Polygons (Saddle Polygons)." §2.2
  • Coxeter, H.S.M.; Regular complex polytopes (1974). Chapter 1. Regular polygons, 1.5. Regular polygons in n dimensions, 1.7. Zigzag and antiprismatic polygons, 1.8. Helical polygons. 4.3. Flags and Orthoschemes, 11.3. Petrie polygons
  • Coxeter, H. S. M. Petrie Polygons. Regular Polytopes, 3rd ed. New York: Dover, 1973. (sec 2.6 Petrie Polygons pp. 24–25, and Chapter 12, pp. 213–235, The generalized Petrie polygon )
  • Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.  (1st ed, 1957) 5.2 The Petrie polygon {p,q}.
  • John Milnor: On the total curvature of knots, Ann. Math. 52 (1950) 248–257.
  • J.M. Sullivan: Curves of finite total curvature, ArXiv:math.0606007v2

External links[edit]