In geometry, a
2 The Petrie polygons of the regular polyhedra
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.
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
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
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 :
5-cell 5 sides V:(5,0)
16-cell 8 sides V:(8,0)
tesseract 8 sides V:(8,8,0)
24-cell 12 sides V:(12,6,6,0)
120-cell 30 sides V:((30,60)3,603,30,60,0)
600-cell 30 sides V:(30,30,30,30,0)
Table of irreducible polytope families
Family n n-simplex n-hypercube n-orthoplex n-demicube 1k2 2k1 k21 pentagonal polytope
Group An Bn
E6 E7 E8 F4 G2
p-gon (example: p=7)
^ 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,
^ Gorini, Catherine A. (2000),
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
Weisstein, Eric W. "Petrie polygon". MathWorld.
Weisstein, Eric W. "
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 p-gon Hexagon Pentagon
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope