Petrie polygon
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides (but no n) belongs to one of the facets. 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 side (but no three) belongs to one of the faces.[1]
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 higherdimensional regular polytopes.
Petrie polygons can be defined more generally for any embedded graph. They form the faces of another embedding of the same graph, usually on a different surface, called the Petrie dual.[2]
History
John Flinders Petrie (1907–1972) was the only son of Egyptologist 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 fourdimensional 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.[3]
In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H.T. Flather to produce The FiftyNine Icosahedra for publication.[4] Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes.
The idea of Petrie polygons was later extended to semiregular polytopes.
The Petrie polygons of the regular polyhedra
The regular duals, {p,q} and {q,p}, are contained within the same projected Petrie polygon. In the images of dual compounds on the right it can be seen that their Petrie polygons have rectangular intersections in the points where the edges touch the common midsphere.
Square  Hexagon  Decagon  

tetrahedron {3,3}  cube {4,3}  octahedron {3,4}  dodecahedron {5,3}  icosahedron {3,5} 
edgecentered  vertexcentered  facecentered  facecentered  vertexcentered 
V:(4,0)  V:(6,2)  V:(6,0)  V:(10,10,0)  V:(10,2) 
The Petrie polygons are the exterior of these orthogonal projections. 
The Petrie polygons of the Kepler–Poinsot polyhedra are hexagons {6} and decagrams {10/3}.
Hexagon  Decagram  

gD {5,5/2}  sD {5,5/2}  gI {3,5/2}  gsD {5/2,3} 
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.
Infinite regular skew polygons also exist as Petrie polygons of the regular hyperbolic tilings, like the order7 triangular tiling, {3,7}:
The Petrie polygon of regular polychora (4polytopes)
The Petrie polygon for the regular polychora {p, q ,r} can also be determined.
{3,3,3} 5cell 5 sides V:(5,0) 
{3,3,4} 16cell 8 sides V:(8,0) 
{4,3,3} tesseract 8 sides V:(8,8,0) 
{3,4,3} 24cell 12 sides V:(12,6,6,0) 
{5,3,3} 120cell 30 sides V:((30,60)^{3},60^{3},30,60,0) 
{3,3,5} 600cell 30 sides V:(30,30,30,30,0) 
The Petrie polygon projections of regular and uniform polytopes
The Petrie polygon projections are useful for the visualization of polytopes of dimension four and higher.
Hypercubes
A hypercube of dimension n has a Petrie polygon of size 2n, which is also the number of its facets.
So each of the (n−1)cubes forming its surface has n−1 sides of the Petrie polygon among its edges.
Hypercubes  

The 1cubes's Petrie digon looks identical to the 1cube. But the 1cube has a single edge, while the digon has two.
(For n=1 the first and the second half are the two distinct but coinciding edges of a digon.)
 
Square  Cube  Tesseract 
Irreducible polytope families
This table represents Petrie polygon projections of 3 regular families (simplex, hypercube, orthoplex), and the exceptional Lie group E_{n} which generate semiregular and uniform polytopes for dimensions 4 to 8.
Table of irreducible polytope families  

Family n 
nsimplex  nhypercube  northoplex  ndemicube  1_{k2}  2_{k1}  k_{21}  pentagonal polytope  
Group  A_{n}  B_{n} 


H_{n}  
2  pgon (example: p=7) 
Hexagon 
Pentagon  
3  Tetrahedron 
Cube 
Octahedron 
Tetrahedron 
Dodecahedron 
Icosahedron  
4  5cell 
16cell 
24cell 
120cell 
600cell  
5  5simplex 
5cube 
5orthoplex 
5demicube 

6  6simplex 
6cube 
6orthoplex 
6demicube 
1_{22} 
2_{21} 

7  7simplex 
7cube 
7orthoplex 
7demicube 
1_{32} 
2_{31} 
3_{21} 

8  8simplex 
8cube 
8orthoplex 
8demicube 
1_{42} 
2_{41} 
4_{21} 

9  9simplex 
9cube 
9orthoplex 
9demicube 

10  10simplex 
10cube 
10orthoplex 
10demicube 
Notes
 Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 (Definition: paper 13, Discrete groups generated by reflections, 1933, p. 161)
 Gorini, Catherine A. (2000), Geometry at Work, MAA Notes, 53, Cambridge University Press, p. 181, ISBN 9780883851647
 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
 H. S. M. Coxeter, Patrick du Val, H.T. Flather, J.F. Petrie (1938) The Fiftynine Icosahedra, University of Toronto studies, mathematical series 6: 1–26
 http://cms.math.ca/openaccess/cjm/v10/cjm1958v10.02200221.pdf
References
 Coxeter, H. S. M. (1947, 63, 73) 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. (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. 135)
 Coxeter, H. S. M. (1999) The Beauty of Geometry: Twelve Essays, Dover Publications LCCN 9935678
 Peter McMullen, Egon Schulte (2002) Abstract Regular Polytopes, Cambridge University Press. ISBN 0521814960
 Steinberg, Robert,ON THE NUMBER OF SIDES OF A PETRIE POLYGON
See also
External links
Wikimedia Commons has media related to Petrie polygons. 
 Weisstein, Eric W. "Petrie polygon". MathWorld.
 Weisstein, Eric W. "Hypercube graphs". MathWorld.
 Weisstein, Eric W. "Cross polytope graphs". MathWorld.
 Weisstein, Eric W. "24cell graph". MathWorld.
 Weisstein, Eric W. "120cell graph". MathWorld.
 Weisstein, Eric W. "600cell graph". MathWorld.
 Weisstein, Eric W. "Gosset graph 3_21". MathWorld.