In geometry , a PETRIE POLYGON for a regular polytope of n dimensions is a skew polygon such that every (n1) 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
CONTENTS * 1 History
* 2 The Petrie polygons of the regular polyhedra
* 3 The
Petrie polygon
HISTORY The
Petrie polygon
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
In 1938 Petrie collaborated with Coxeter,
Patrick du Val , and H.T.
Flather to produce
The FiftyNine Icosahedra for publication.
Realizing the geometric facility of the skew polygons used by Petrie,
Coxeter
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
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 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. 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 order7 triangular tiling , {3,7}: THE PETRIE POLYGON OF REGULAR POLYCHORA (4POLYTOPES) The
Petrie polygon
{3,3,3} 5cell 5 sides V:(5,0) {3,3,4}
16cell
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)3,603,30,60,0) {3,3,5} 600cell 30 sides V:(30,30,30,30,0) THE PETRIE POLYGON PROJECTIONS OF REGULAR AND UNIFORM POLYTOPES The
Petrie polygon
TABLE OF IRREDUCIBLE POLYTOPE FAMILIES Family n NSIMPLEX NHYPERCUBE NORTHOPLEX NDEMICUBE 1K2 2K1 K21 PENTAGONAL POLYTOPE GROUP AN BCN I2(p) Dn E6 E7 E8 F4 G2 HN 2 Triangle pgon (example: p=7 ) 3 4 5 6 122 221 7 132 231 321 8 142 241 421 9 10 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)
* ^ H.S.M.
Coxeter
REFERENCES *
Coxeter
