In geometry, a
Petrie polygon
Contents 1 History
2 The Petrie polygons of the regular polyhedra
3 The
Petrie polygon
History[edit] 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
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
h + 2 = 24/(10 − p − q).[5] 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
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
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)3,603,30,60,0) 3,3,5 600cell 30 sides V:(30,30,30,30,0) The
Petrie polygon
Table of irreducible polytope families Family n nsimplex nhypercube northoplex ndemicube 1k2 2k1 k21 pentagonal polytope Group An Bn I2(p) Dn E6 E7 E8 F4 G2 Hn 2 Triangle Square pgon (example: p=7) Hexagon Pentagon 3 Tetrahedron Cube Octahedron Tetrahedron Dodecahedron Icosahedron 4 5cell Tesseract 16cell Demitesseract 24cell 120cell 600cell 5 5simplex 5cube 5orthoplex 5demicube 6 6simplex 6cube 6orthoplex 6demicube 122 221 7 7simplex 7cube 7orthoplex 7demicube 132 231 321 8 8simplex 8cube 8orthoplex 8demicube 142 241 421 9 9simplex 9cube 9orthoplex 9demicube 10 10simplex 10cube 10orthoplex 10demicube Notes[edit] ^ 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 [1]
(Definition: paper 13, Discrete groups generated by reflections, 1933,
p. 161)
^ Gorini, Catherine A. (2000),
Geometry
References[edit] 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
External links[edit] Weisstein, Eric W. "Petrie polygon". MathWorld.
Weisstein, Eric W. "
Hypercube
v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square pgon Hexagon Pentagon Uniform polyhedron
Tetrahedron
Octahedron
Dodecahedron
Uniform 4polytope
5cell
16cell
Uniform 5polytope
5simplex
5orthoplex
Uniform 6polytope
6simplex
6orthoplex
Uniform 7polytope
7simplex
7orthoplex
Uniform 8polytope
8simplex
8orthoplex
Uniform 9polytope
9simplex
9orthoplex
Uniform 10polytope
10simplex
10orthoplex
Uniform npolytope nsimplex northoplex • ncube ndemicube 1k2 • 2k1 • k21 npentagonal polytope Topics:
Polytope
