View source for Petrie polygon

{{Icosahedron visualizations}}

In [[geometry]], a '''Petrie polygon''' for a [[regular polytope]] of ''n'' dimensions is a [[skew polygon]] such that every (''n''-1) consecutive [[Edge (geometry)|sides]] (but no ''n'') belong to one of the [[Facet (geometry)|facet]]s.  The '''Petrie polygon''' of a [[regular polygon]] is the regular polygon itself; that of a [[regular polyhedron]] is a [[skew polygon]] such that every two consecutive [[Edge (geometry)|side]] (but no three) belongs to one of the [[Face (geometry)|face]]s.<ref>'''Kaleidoscopes: Selected Writings of H. S. M. Coxeter''', edited by F. Arthur Sherk, [[Peter McMullen]], Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] (Definition: paper 13, Discrete groups generated by reflections, 1933, p. 161)</ref>

For every regular polytope there exists an [[orthogonal projection]] onto a plane such that one Petrie polygon becomes a [[regular polygon]] with the remainder of the projection interior to it. The plane in question is the [[Coxeter plane]] of the [[symmetry group]] of the polygon, and the number of sides, ''h,'' is [[Coxeter number]] of the [[Coxeter group]]. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes.

== History ==
[[File:Petrie_polygon_mapping_cube_to_hexagon.png|thumb|The ''Petrie polygon'' for a [[cube]] is a skew hexagon passing through 6 of 8 vertices. The skew ''Petrie polygon'' can be seen as a regular planar polygon by a specific [[orthogonal projection]].]]

John Flinders Petrie (1907–1972) was the only son of [[Egyptologist]] [[Sir W. M. Flinders Petrie|Flinders Petrie]]. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability. In periods of intense concentration he could answer questions about complicated four-dimensional objects by ''visualizing'' them.

He first noted the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. 
Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra:
:One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedral; infinite but free of false vertices. When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex, and one consisting of hexagons, four at each vertex.<ref>H.S.M. Coxeter (1937) "Regular skew polyhedral in three and four dimensions and their topological analogues", [[Proceedings of the London Mathematical Society]] (2) 43: 33 to 62</ref>
In 1938 Petrie collaborated with Coxeter, [[Patrick du Val]], and H.T. Flather to produce ''[[The Fifty-Nine Icosahedra]]'' for publication.<ref>H. S. M. Coxeter, [[Patrick du Val]], H.T. Flather, J.F. Petrie (1938) ''The Fifty-nine Icosahedra'', [[University of Toronto]] studies, mathematical series 6: 1–26</ref>
Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote [[Regular Polytopes]].

In 1972, a few months after his retirement, Petrie was killed by a car while attempting to cross a motorway near his home in [[Surrey]].<ref>H.S.M. Coxeter (1973) [[Regular Polytopes (book)|Regular Polytopes]], 3rd edition, page 32</ref>

The idea of Petrie polygons was later extended to [[semiregular polytope]]s.

== The Petrie polygons of the regular polyhedra ==
The Petrie polygon of the regular polyhedron {''p'',&nbsp;''q''} has ''h'' sides, where 
:h+2=24/(10-p-q).<ref>http://cms.math.ca/openaccess/cjm/v10/cjm1958v10.0220-0221.pdf</ref>

The [[dual polyhedron|regular duals]], {p,q} and {q,p}, are contained within the same projected Petrie polygon.

{| class=wikitable width=720px 
|+ Petrie polygons for regular polyhedra (red polygons)
|- align=center
!Square||colspan=2|Hexagon||colspan=2|Decagon
|- align=center
|colspan=5|[[Image:Petrie polygons.png|720px]]
|- align=center
|[[tetrahedron]]||[[cube]]||[[octahedron]]||[[dodecahedron]]||[[icosahedron]]
|- align=center
|{{CDD|node_1|3|node|3|node}}
|{{CDD|node_1|4|node|3|node}}
|{{CDD|node_1|3|node|4|node}}
|{{CDD|node_1|5|node|3|node}}
|{{CDD|node_1|3|node|5|node}}
|- align=center
|edge-centered||vertex-centered||face-centered||face-centered||vertex-centered
|- align=center
|''V'':(4,0)||''V'':(6,2)||''V'':(6,0)||''V'':(10,10,0)||''V'':(10,2)
|-
|colspan=5|The Petrie polygons are the exterior of these orthogonal projections. Blue show "front" edges, while black lines show back edges.<BR>
The concentric rings of vertices are counted starting from the outside working inwards with a notation: ''V'':(''a'',&nbsp;''b'',&nbsp;...), ending in zero if there are no central vertices.
|}

Three of the [[Kepler–Poinsot polyhedra]] have [[hexagon]]al, {6}, and [[decagram (geometry)|decagram]]mic, {10/3}, petrie polygons.
{| class=wikitable
|+ Kepler–Poinsot polyhedra
|- align=center 
!colspan=2|Hexagon||colspan=2|Decagram
|- align=center 
|[[File:Small_stellated_dodecahedron-petrie.png|160px]]
|[[File:Great_dodecahedron-petrie.png|160px]]
|[[File:Great_stellated_dodecahedron_petrie.png|160px]]
|[[File:Great_icosahedron_petrie.png|160px]]
|- align=center 
|[[Small_stellated_dodecahedron|{5,5/2}]]||[[Great_dodecahedron|{5,5/2}]]||[[Great_stellated_dodecahedron|{5/2,3}]]||[[Great_icosahedron|{3,5/2}]]
|- align=center 
|{{CDD|node_1|5|node|5-2|node}}||{{CDD|node_1|5|node|5-2|node}}||{{CDD|node_1|5-2|node|3|node}}||{{CDD|node_1|3|node|5-2|node}}
|}
Infinite regular skew polygons ([[apeirogon]]) can also be defined as being the Petrie polygons of the regular tilings, having angles of 90, 120, and 60 degrees of their square, hexagon and triangular faces respectively.
:[[File:petrie_polygons_of_regular_tilings.png|720px]]

Infinite regular skew polygons also exist as Petrie polygons of the regular hyperbolic tilings, like the [[order-7 triangular tiling]], {3,7}:
:[[File:Order-7 triangular tiling petrie polygon.png|240px]]

== The Petrie polygon of regular polychora (4-polytopes) ==

The Petrie polygon for the regular polychora {''p'',&nbsp;''q''&nbsp;,''r''} can also be determined.

{| class=wikitable width=480
|- align=center
|[[Image:4-simplex t0.svg|160px]]<BR>{3,3,3}<BR>{{CDD|node_1|3|node|3|node|3|node}}<BR>[[5-cell]]<BR>5 sides<BR>''V'':(5,0)
|[[Image:4-orthoplex.svg|160px]]<BR>{3,3,4}<BR>{{CDD|node_1|3|node|3|node|4|node}}<BR>[[16-cell]]<BR>8 sides<BR>''V'':(8,0)
|align=center|[[File:4-cube graph.svg|160px]]<BR>{4,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node}}<BR>[[tesseract]]<BR>8 sides<BR>''V'':(8,8,0)
|- align=center
|[[File:24-cell t0 F4.svg|160px]]<BR>{3,4,3}<BR>{{CDD|node_1|3|node|4|node|3|node}}<BR>[[24-cell]]<BR>12 sides<BR>''V'':(12,6,6,0)
|[[Image:120-cell graph H4.svg|160px]]<BR>{5,3,3}<BR>{{CDD|node_1|5|node|3|node|3|node}}<BR>[[120-cell]]<BR>30 sides<BR>''V'':((30,60)<sup>3</sup>,60<sup>3</sup>,30,60,0)
|[[File:600-cell graph H4.svg|160px]]<BR>{3,3,5}<BR>{{CDD|node_1|3|node|3|node|5|node}}<BR>[[600-cell]]<BR>30 sides<BR>V:(30,30,30,30,0)
|}

== The Petrie polygon projections of regular and uniform polytopes ==

The Petrie polygon projections are most useful for visualization of polytopes of dimension four and higher. This table represents Petrie polygon projections of 3 regular families ([[simplex]], [[hypercube]], [[orthoplex]]), and the [[exceptional Lie group]] E<sub>n</sub> which generate semiregular and uniform polytopes for dimensions 4 to 8.
{{:Polytope families}}

== Notes ==
{{reflist}}

== References ==
* [[Coxeter]], H. S. M. (1947, 63, 73) [[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 '')
* Coxeter, H.S.M. (1974) ''Regular complex polytopes''. Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons
* Ball, W. W. R. and H. S. M. Coxeter (1987) ''Mathematical Recreations and Essays'', 13th ed. New York: Dover. (p.&nbsp;135)
* Coxeter, H. S. M. (1999) ''The Beauty of Geometry: Twelve Essays'', Dover Publications {{LCCN|99035678}}
* [[Peter McMullen]], Egon Schulte (2002) ''Abstract Regular Polytopes'', [[Cambridge University Press]]. ISBN 0-521-81496-0
* Steinberg, Robert,''ON THE NUMBER OF SIDES OF A PETRIE POLYGON''

== External links==
* {{mathworld | urlname = PetriePolygon | title = Petrie polygon}}
* {{mathworld | urlname = HypercubeGraph | title = Hypercube graphs}}
* {{mathworld | urlname = CrossPolytope | title = Cross polytope graphs}}
* {{mathworld | urlname = 24-Cell | title = 24-cell graph}}
* {{mathworld | urlname = 120-Cell | title = 120-cell graph}}
* {{mathworld | urlname = 600-Cell | title = 600-cell graph}}
* {{mathworld | urlname = GossetGraph | title = Gosset graph 3_21}}
{{Polytopes}}

{{DEFAULTSORT:Petrie Polygon}}
[[Category:Polytopes]]