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Petrie polygon

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Petrie polygon

Various visualizations of the icosahedron

2D

Net

Petrie

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive sides (but no n) belong 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 sides (but no three) belong 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 higher-dimensional regular polytopes.

History

The Petrie polygon for a cube is a skew hexagon passing through 6 of 8 vertices. The skew Petrie polygon can be seen as an regular planer polygon by a specific orthogonal projection.

John Flinders Petrie (1907-1972) was the only son of Egyptologist Sir W. M. 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 realized the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. He was a lifelong friend of Coxeter, who named these polygons after him.

The idea of Petrie polygons was later extended to semiregular 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.

The Petrie polygons of the regular polyhedra

The Petrie polygon of the regular polyhedron {pq} has h sides, where

cos2(π/h) = cos2(π/p) + cos2(π/q).

The regular duals, {p,q} and {q,p}, are contained within the same projected Petrie polygon.

Petrie polygons for regular polyhedra (red polygons)
tetrahedron cube octahedron dodecahedron icosahedron
edge-centered vertex-centered face-centered face-centered vertex-centered
4 sides 6 sides 6 sides 10 sides 10 sides
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. Blue show "front" edges, while black lines show back edges.

The concentric rings of vertices are counted starting from the outside working inwards with a notation: V:(ab, ...), ending in zero if there are no central vertices.

The Petrie polygon of regular polychora (4-polytopes)

The Petrie polygon for the regular polychora {pq ,r} can also be determined.


{3,3,3}

5-cell
5 sides
V:(5,0)

{3,3,4}

16-cell
8 sides
V:(8,0)

{4,3,3}

tesseract
8 sides
V:(8,8,0)

{3,4,3}

24-cell
12 sides
V:(12,6,6,0)

{5,3,3}

120-cell
30 sides
V:((30,60)3,603,30,60,0)

{3,3,5}

600-cell
30 sides
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 En which generate semiregular and uniform polytopes for dimensions 4 to 8.


Table of irreducible polytope families
Family
n
n-simplex n-hypercube n-orthoplex n-demicube 1k2 2k1 k21 pentagonal polytope
Group An BCn
I2(p) Dn
E6 E7 E8 F4 G2
Hn
2

(Triangle)


Square


p-gon


Hexagon


Pentagon
3

Tetrahedron


Cube


Octahedron


Tetrahedron
 

Dodecahedron


Icosahedron
4

5-cell

Tesseract



16-cell

Demitesseract



24-cell


120-cell


600-cell
5

5-simplex


5-cube


5-orthoplex


5-demicube
   
6

6-simplex


6-cube


6-orthoplex


6-demicube


122


221
 
7

7-simplex


7-cube


7-orthoplex


7-demicube


132


231


321
 
8

8-simplex


8-cube


8-orthoplex


8-demicube


142


241


421
 
9

9-simplex


9-cube


9-orthoplex


9-demicube
 
10

10-simplex


10-cube


10-orthoplex


10-demicube
 


Notes

  1. ^ 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 [1] (Definition: paper 13, Discrete groups generated by reflections, 1933, p. 161)

References

  • Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
  • Coxeter, H. S. M. The Beauty of Geometry: Twelve Essays (1999), Dover Publications ISBN 99-35678
  • Coxeter, H.S.M.; Regular complex polytopes (1974). Section 4.3 Flags and Orthoschemes, Section 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.; Regular complex polytopes (1974).
  • Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987. (p. 135)

External links

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