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The Info List - Petrie Polygon


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In geometry , a PETRIE POLYGON for a regular polytope of _n_ dimensions is a skew polygon such that every (_n_-1) consecutive side (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 .

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.

CONTENTS

* 1 History * 2 The Petrie polygons of the regular polyhedra * 3 The Petrie polygon of regular polychora (4-polytopes) * 4 The Petrie polygon projections of regular and uniform polytopes * 5 Notes * 6 References * 7 External links

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 a regular planar polygon by a specific orthogonal projection .

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.

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

The idea of Petrie polygons was later extended to semiregular polytopes .

THE PETRIE POLYGONS OF THE REGULAR POLYHEDRA

The Petrie polygon of the regular polyhedron {_p_, _q_} has _h_ sides, where h+2=24/(10-p-q).

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

Petrie polygons for regular polyhedra (red polygons) SQUARE HEXAGON DECAGON

_

tetrahedron cube octahedron dodecahedron icosahedron

edge-centered vertex-centered face-centered face-centered vertex-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. 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_:(_a_, _b_, ...), ending in zero if there are no central vertices.

Three of the Kepler–Poinsot polyhedra have hexagonal , {6}, and decagrammic , {10/3}, petrie polygons.

Kepler–Poinsot polyhedra HEXAGON DECAGRAM

{5,5/2} {5,5/2} {5/2,3} {3,5/2}

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 order-7 triangular tiling , {3,7}:

THE PETRIE POLYGON OF REGULAR POLYCHORA (4-POLYTOPES)

The Petrie polygon for the regular polychora {_p_, _q_ ,_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 (example: p=7 )

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