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In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) 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. 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, [1] (Definition: paper 13, Discrete groups generated by reflections, 1933, p. 161) Petrie polygons are named for mathematician John Flinders Petrie.
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, , 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 Graph embedding. They form the faces of another embedding of the same graph, usually on a different surface, called the Petrie dual.. See in particular p. 181.
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:
The idea of Petrie polygons was later extended to semiregular polytopes.
| + Petrie polygons for Platonic solids | ||||
| tetrahedron {3,3} | cube {4,3} | octahedron {3,4} | dodecahedron {5,3} | icosahedron {3,5} |
| vertex-centered | ||||
| 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. |
The Petrie polygons of the Kepler–Poinsot polyhedra are {6} and decagrams {10/3}.
| + Petrie polygons for Kepler–Poinsot polyhedra | |||
| gD {5,5/2} | sD {5,5/2} | gI {3,5/2} | gsD {5/2,3} |
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}:
{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) | {3,3,5} 600-cell 30 sides V:(30,30,30,30,0) | {5,3,3} 120-cell 30 sides V:((30,60)3,603,30,60,0) |
| The 1-cubes's Petrie digon looks identical to the 1-cube. But the 1-cube has a single edge, while the digon has two. The 2-cube's Petrie square is identical to the 2-cube. Each pair of consecutive sides of the 3-cube's Petrie hexagon belongs to one of its six square faces. Each triple of consecutive sides of the 4-cube's Petrie octagon belongs to one of its eight cube cells. The images show how the Petrie polygon for dimension n + 1 can be constructed from that for dimension n:
The sides of each Petrie polygon belong to these dimensions: | ||
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