Ingeometry, aPetrie polygon for aregular polytope ofn dimensions is askew polygon in which everyn – 1 consecutivesides (but non) belongs to one of thefacets. ThePetrie polygon of aregular polygon is the regular polygon itself; that of aregular polyhedron is a skew polygon such that every two consecutive sides (but no three) belongs to one of thefaces.[1] Petrie polygons are named for mathematicianJohn Flinders Petrie.
For every regular polytope there exists anorthogonal 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 theCoxeter plane of thesymmetry group of the polygon, and the number of sides,h, is theCoxeter number of theCoxeter 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 anyembedded graph. They form the faces of another embedding of the same graph, usually on a different surface, called thePetrie dual.[2]
John Flinders Petrie (1907–1972) was the son ofEgyptologistsHilda andFlinders 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 byvisualizing 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:
In 1938 Petrie collaborated with Coxeter,Patrick du Val, and H. T. Flather to produceThe Fifty-Nine Icosahedra for publication.[4]Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wroteRegular Polytopes.
The idea of Petrie polygons was later extended tosemiregular polytopes.
Theregular duals, {p,q} and {q,p}, are contained within the same projected Petrie polygon.In the images ofdual compounds on the right it can be seen that their Petrie polygons have rectangular intersections in the points where the edges touch the commonmidsphere.
| Square | Hexagon | Decagon | ||
|---|---|---|---|---|
 |  |  |  |  |
| tetrahedron {3,3} | cube {4,3} | octahedron {3,4} | dodecahedron {5,3} | icosahedron {3,5} |
   |  | |  | |
| 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. | ||||
The Petrie polygons of theKepler–Poinsot polyhedra arehexagons {6} anddecagrams {10/3}.
| Hexagon | Decagram | ||
|---|---|---|---|
 |  |  |  |
| 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 theorder-7 triangular tiling, {3,7}:
The Petrie polygon for the regular polychora {p, q ,r} can also be determined, such that every three consecutive sides (but no four) belong to one of the polychoron's cells. As the surface of a 4-polytope is a 3-dimensional space (the3-sphere), the Petrie polygon of a regular 4-polytope is a 3-dimensional helix in this surface.
 {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 Petrie polygon projections are useful for the visualization of polytopes of dimension four and higher.
Ahypercube of dimensionn has a Petrie polygon of size 2n, which is also the number of itsfacets.
So each of the (n − 1)-cubes forming itssurface hasn − 1 sides of the Petrie polygon among its edges.
| Hypercubes | ||
|---|---|---|
The 1-cubes's Petriedigon looks identical to the 1-cube. But the 1-cube has a single edge, while the digon has two. The images show how the Petrie polygon for dimensionn + 1 can be constructed from that for dimension n:
(Forn = 1 the first and the second half are the two distinct but coinciding edges of a digon.) The sides of each Petrie polygon belong to these dimensions: | ||
| Square | Cube | Tesseract |
 |  |  |
 |  |  |
