View source for Skew polygon

{{short description|Polygonal chain whose vertices are not all coplanar}}
{{Inline|date=January 2024}}
[[File:Disphenoid tetrahedron.png|260px|thumb|The red edges of this [[tetragonal disphenoid]] represent a regular zig-zag skew quadrilateral.]]

In [[geometry]], a '''skew polygon''' is a closed [[polygonal chain]] in [[Euclidean space]]. It is a [[figure (geometry)|figure]] similar to a [[polygon]] except its [[Vertex (geometry)|vertices]] are not all [[coplanarity|coplanar]].{{Sfn|Coxeter|1973|loc=§1.1 Regular polygons|ps=; "If the vertices are all coplanar, we speak of a ''plane'' polygon, otherwise a ''skew'' polygon."}}  While a polygon is ordinarily defined as a [[plane figure]], the [[edge (geometry)|edges]] and vertices of a skew polygon form a [[space curve]]. Skew polygons must have at least four vertices. The ''interior'' [[surface (geometry)|surface]] and corresponding area measure of such a polygon is not uniquely defined.

[[Skew infinite polygon]]s (apeirogons) have vertices which are not all colinear.

A '''zig-zag skew polygon''' or '''[[antiprism]]atic polygon'''<ref name=rcp>Regular complex polytopes, p. 6</ref> 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.

==Skew polygons in three dimensions==
[[File:Dodecagonal antiprism.png|160px|thumb|A uniform ''n''-gonal [[antiprism]] has a 2''n''-sided regular skew polygon defined along its side edges.]]
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 {{mvar|n}}-gon can be given a [[Schläfli symbol]] {{math|{{(}}{{mvar|p}}{{)}}#{{(}}{{)}}}} as a ''blend'' of a [[regular polygon]] {{math|{{{mvar|p}}}}} and an orthogonal [[line segment]] { }.<ref>Abstract Regular Polytopes, p.217</ref> The symmetry operation between sequential vertices is [[glide reflection]].

Examples are shown on the uniform square and pentagon antiprisms. The [[star antiprism]]s 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.

{| class=wikitable style="text-align:center;"
|+ Regular zig-zag skew polygons
|-
| Skew square
| [[Skew hexagon]]
| [[Skew octagon]]
| colspan=3 | [[Skew decagon]]
| [[Skew dodecagon]]
|-
| {4}#{ }
| [[hexagon#skew hexagon|{6}#{ }]]
| [[octagon#skew octagon|{8}#{ }]]
| [[decagon#skew decagon|{10}#{ }]]
| {5}#{ }
| {5/2}#{ }
| [[dodecagon#skew dodecagon|{12}#{ }]]
|-
| [[File:Disphenoid tetrahedron.png|120px]]
| [[File:Skew_polygon_in_triangular_antiprism.png|100px]]
| [[File:Skew polygon in square antiprism.png|100px]]
| [[File:Regular skew polygon in pentagonal antiprism.png|100px]]
| [[File:Regular skew polygon in pentagrammic antiprism.png|100px]]
| [[File:Regular skew polygon in pentagrammic crossed-antiprism.png|100px]]
| [[File:Regular skew polygon in hexagonal antiprism.png|100px]]
|-
| [[tetrahedron|s{2,4}]]
| [[octahedron|s{2,6}]]
| [[Square antiprism|s{2,8}]]
| [[Pentagonal antiprism|s{2,10}]]
| [[pentagrammic antiprism|sr{2,5/2}]]
| [[pentagrammic crossed-antiprism|s{2,10/3}]]
| [[hexagonal antiprism|s{2,12}]]
|}

[[Petrie polygon]]s are regular skew polygons defined within regular polyhedra and polytopes. For example, the five [[Platonic solid]]s have 4-, 6-, and 10-sided regular skew polygons, as seen in these [[orthogonal projection]]s with red edges around their respective [[projective envelope|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.

{{multiple image
 | align = left  | total_width = 350
 | image1 = Skeleton pair 4-4, Petrie, stick, size m.png
 | image2 = Skeleton pair 6-8, Petrie, stick, size m.png
 | image3 = Skeleton pair 12-20, Petrie, stick, size m.png
 | footer = Petries polygons of Platonic solids
}}
{{clear}}

== Regular skew polygon as vertex figure of regular skew polyhedron ==
A [[regular skew polyhedron]] has regular polygon faces, and a regular skew polygon [[vertex figures|vertex figure]].

Three infinite regular skew polyhedra are [[regular skew apeirohedron|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-polytope|uniform 4-polytopes]].

{| class=wikitable style="text-align:center;"
|+ Skew [[vertex figure]]s of the 3 infinite regular skew polyhedra
! {4,6{{pipe}}4}
! {6,4{{pipe}}4}
! {6,6{{pipe}}3}
|- valign=top
| [[File:Six-square skew polyhedron-vf.png|160px]]<br>Regular skew hexagon<br>{3}#{ }
| [[File:Four-hexagon skew polyhedron-vf.png|160px]]<br>Regular skew square<br>{2}#{ }
| [[File:Six-hexagon skew polyhedron-vf.png|160px]]<br>Regular skew hexagon<br>{3}#{ }
|}

==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 polygon]]s of the [[regular 4-polytope|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.

{| class=wikitable width=480
!A<sub>4</sub>, [3,3,3]
!colspan=2|B<sub>4</sub>, [4,3,3]
!F<sub>4</sub>, [3,4,3]
!colspan=2|H<sub>4</sub>, [5,3,3]
|-
![[Pentagon#Graphs|Pentagon]]
!colspan=2|[[Octagon#Petrie polygons|Octagon]]
![[Dodecagon#Petrie polygons|Dodecagon]]
!colspan=2|[[Triacontagon#Petrie polygons|Triacontagon]]
|- align=center
|[[File:4-simplex t0.svg|80px]]<br>[[5-cell]]<br>{3,3,3}
|[[File:4-cube graph.svg|80px]]<br>[[tesseract]]<br>{4,3,3}
|[[File:4-orthoplex.svg|80px]]<br>[[16-cell]]<br>{3,3,4}
|[[File:24-cell t0 F4.svg|80px]]<br>[[24-cell]]<br>{3,4,3}
|[[File:120-cell graph H4.svg|80px]]<br>[[120-cell]]<br>{5,3,3}
|[[File:600-cell graph H4.svg|80px]]<br>[[600-cell]]<br>{3,3,5}
|}

The ''n''-''n'' [[duoprism|duoprisms]] and dual [[duopyramid]]s also have 2''n''-gonal Petrie polygons. (The [[tesseract]] is a 4-4 duoprism, and the [[16-cell]] is a 4-4 duopyramid.)
{| class=wikitable width=480
!colspan=2|[[Hexagon#Petrie polygons|Hexagon]]
!colspan=2|[[Decagon#Petrie polygons|Decagon]]
!colspan=2|[[Dodecagon#Petrie polygons|Dodecagon]]
|- align=center
|[[File:3-3 duoprism ortho-Dih3.png|80px]]<br>[[3-3 duoprism]]
|[[File:3-3 duopyramid ortho.png|80px]]<br>[[3-3 duopyramid]]
|[[File:5-5 duoprism ortho-Dih5.png|80px]]<br>[[5-5 duoprism]]
|[[File:5-5 duopyramid ortho.png|80px]]<br>[[5-5 duopyramid]]
|[[File:6-6 duoprism ortho-3.png|80px]]<br>[[6-6 duoprism]]
|[[File:6-6 duopyramid ortho-3.png|80px]]<br>[[6-6 duopyramid]]
|}

==See also==
*[[Petrie polygon]]
*{{slink|Quadrilateral|Skew quadrilaterals}}
*[[Regular skew polyhedron]]
*[[Skew apeirohedron]] (infinite skew polyhedron)
*[[Skew lines]]

==Citations==
{{Reflist}}

==References==
*{{citation |last1= McMullen |first1= Peter |author1-link= Peter McMullen |first2= Egon |last2= Schulte |title= Abstract Regular Polytopes |edition= 1st |publisher= [[Cambridge University Press]] |isbn= 0-521-81496-0 |date= December 2002 |url-access= registration |url= https://archive.org/details/abstractregularp0000mcmu }} p.&nbsp;25
*{{The Geometrical Foundation of Natural Structure (book)}} "Skew Polygons (Saddle Polygons)" §2.2
* {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
*[[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 (book)|Regular Polytopes]], 3rd ed. New York: Dover, 1973. (sec 2.6 ''Petrie Polygons'' pp.&nbsp;24&ndash;25, and Chapter 12, pp.&nbsp;213–235, ''The generalized Petrie polygon'')
*{{cite book |author1=Coxeter, H. S. M. |author2=Moser, W. O. J.  |name-list-style=amp |title=Generators and Relations for Discrete Groups |location=New York |publisher=Springer-Verlag |year=1980 |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.
*[[John M. Sullivan (mathematician)|J.M. Sullivan]]: ''Curves of finite total curvature'', ArXiv:math.0606007v2

==External links==
*{{MathWorld|title=Skew polygon|urlname=SkewPolygon}}
*{{MathWorld|title=Petrie polygon|urlname=PetriePolygon}}

{{Polygons}}
[[Category:Types of polygons]]