Petrie polygon

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Skeleton 12, Petrie, stick, size l.png
Skeleton 12, Petrie, stick, size l, 5-fold.png
The Petrie polygon of the dodecahedron is a skew decagon. Seen from the solid's 5-fold symmetry axis it looks like a regular decagon. Every pair of consecutive sides belongs to one pentagon (but no triple does).

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 sides (but no three) belongs to one of the faces. [1] Petrie polygons are named for mathematician John Flinders Petrie.

Contents

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 the Coxeter number of the Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional 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 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. [3]

In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H.T. Flather to produce The Fifty-Nine 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

Skeleton pair 4-4, size m, thick.png
Skeleton pair 4-4, Petrie, stick, size m.png
Skeleton pair 4-4, Petrie, stick, size m, 2-fold square.png
Two tetrahedra with Petrie squares
Skeleton pair 6-8, size m, thick.png
Skeleton pair 6-8, Petrie, stick, size m.png
Skeleton pair 6-8, Petrie, stick, size m, 3-fold.png
Cube and octahedron with Petrie hexagons
Skeleton pair 12-20, size m.png
Skeleton pair 12-20, Petrie, stick, size m.png
Skeleton pair 12-20, Petrie, stick, size m, 5-fold.png
Dodecahedron and icosahedron with Petrie decagons

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.

Petrie polygons for Platonic solids
SquareHexagonDecagon
Skeleton 4b, Petrie, stick, size m, 2-fold square.png Skeleton 6, Petrie, stick, size m, 3-fold.png Skeleton 8, Petrie, stick, size m, 3-fold.png Skeleton 12, Petrie, stick, size m, 5-fold.png Skeleton 20, Petrie, stick, size m, 5-fold.png
tetrahedron {3,3} cube {4,3} octahedron {3,4} dodecahedron {5,3} icosahedron {3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
edge-centeredvertex-centeredface-centeredface-centeredvertex-centered
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 concentric rings of vertices are counted starting from the outside working inwards with a notation: V:(a, b, ...), ending in zero if there are no central vertices.
The number of sides for {p, q} is 24/(10  p  q)  2. [5]

Skeleton pair Gr12 and dual, size s.png
Skeleton pair Gr12 and dual, Petrie, stick, size s.png
Skeleton pair Gr12 and dual, Petrie, stick, size s, 3-fold.png
gD and sD with Petrie hexagons
Skeleton pair Gr20 and dual, size s.png
Skeleton pair Gr20 and dual, Petrie, stick, size s.png
Skeleton pair Gr20 and dual, Petrie, stick, size s, 5-fold.png
gI and gsD with Petrie decagrams

The Petrie polygons of the Kepler–Poinsot polyhedra are hexagons {6} and decagrams {10/3}.

Petrie polygons for Kepler–Poinsot polyhedra
HexagonDecagram
Skeleton Gr12, Petrie, stick, size m, 3-fold.png Skeleton St12, Petrie, stick, size m, 3-fold.png Skeleton Gr20, Petrie, stick, size m, 5-fold.png Skeleton GrSt12, Petrie, stick, size m, 5-fold.png
gD {5,5/2} sD {5,5/2} gI {3,5/2} gsD {5/2,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 5-2.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5-2.pngCDel node.pngCDel node 1.pngCDel 5-2.pngCDel node.pngCDel 3.pngCDel node.png

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.

Petrie polygons of regular tilings.png

Infinite regular skew polygons also exist as Petrie polygons of the regular hyperbolic tilings, like the order-7 triangular tiling, {3,7}:

Order-7 triangular tiling petrie polygon.png

The Petrie polygon of regular polychora (4-polytopes)

Tesseract Schlegel Petrie.png
Numbered 4-cube.svg
The Petrie polygon of the tesseract is an octagon. Every triple of consecutive sides belongs to one of its eight cubic cells.

The Petrie polygon for the regular polychora {p, q ,r} can also be determined, such that every three consecutive sides (but no four) belong to one of the polychoron's cells.

4-simplex t0.svg
{3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cell
5 sides
V:(5,0)
4-orthoplex.svg
{3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
16-cell
8 sides
V:(8,0)
4-cube graph.svg
{4,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
tesseract
8 sides
V:(8,8,0)
24-cell t0 F4.svg
{3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
24-cell
12 sides
V:(12,6,6,0)
600-cell graph H4.svg
{3,3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
600-cell
30 sides
V:(30,30,30,30,0)
120-cell graph H4.svg
{5,3,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell
30 sides
V:((30,60)3,603,30,60,0)

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.

Irreducible polytope families

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 AnBn
I2(p)Dn
E6E7E8F4G2
Hn
2 2-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.png

Triangle

2-cube.svg
CDel node 1.pngCDel 4.pngCDel node.png

Square

Regular polygon 7.svg
CDel node 1.pngCDel p.pngCDel node.png
p-gon
(example: p=7)
Regular polygon 6.svg
CDel node 1.pngCDel 6.pngCDel node.png
Hexagon
Regular polygon 5.svg
CDel node 1.pngCDel 5.pngCDel node.png
Pentagon
3 3-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Tetrahedron
3-cube t0.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cube
3-cube t2.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron
3-demicube.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.png
Tetrahedron
  Dodecahedron H3 projection.svg
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron
Icosahedron H3 projection.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Icosahedron
4 4-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cell
4-cube t0.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Tesseract

4-cube t3.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
16-cell
4-demicube t0 D4.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png

Demitesseract

24-cell t0 F4.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
24-cell
120-cell graph H4.svg
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
120-cell
600-cell graph H4.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
600-cell
5 5-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex
5-cube graph.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube
5-orthoplex.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-orthoplex
5-demicube.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
5-demicube
  
6 6-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex
6-cube graph.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube
6-orthoplex.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-orthoplex
6-demicube.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
6-demicube
Up 1 22 t0 E6.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
122
E6 graph.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
221
 
7 7-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex
7-cube graph.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-cube
7-orthoplex.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-orthoplex
7-demicube.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7-demicube
Gosset 1 32 petrie.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
132
Gosset 2 31 polytope.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
231
E7 graph.svg
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
321
 
8 8-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex
8-cube.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube
8-orthoplex.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-orthoplex
8-demicube.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
8-demicube
Gosset 1 42 polytope petrie.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
142
2 41 polytope petrie.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
241
Gosset 4 21 polytope petrie.svg
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
421
 
9 9-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9-simplex
9-cube.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9-cube
9-orthoplex.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
9-orthoplex
9-demicube.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
9-demicube
 
10 10-simplex t0.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-simplex
10-cube.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube
10-orthoplex.svg
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
10-orthoplex
10-demicube.svg
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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 (Definition: paper 13, Discrete groups generated by reflections, 1933, p. 161)
  2. Gorini, Catherine A. (2000), Geometry at Work, MAA Notes, vol. 53, Cambridge University Press, p. 181, ISBN   9780883851647
  3. 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
  4. 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
  5. http://cms.math.ca/openaccess/cjm/v10/cjm1958v10.0220-0221.pdf [ dead link ]

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The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller.

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.

<span class="mw-page-title-main">Icositetragon</span> Polygon with 24 edges

In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

References

See also

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds