In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides (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 sides (but no three) belongs to one of the faces.^{ [1] } 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, h, is the Coxeter number of the Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higherdimensional regular polytopes.
Petrie polygons can be defined more generally for any embedded graph. They form the faces of another embedding of the same graph, usually on a different surface, called the Petrie dual.^{ [2] }
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 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 produce The FiftyNine Icosahedra for publication.^{ [4] } Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes .
The idea of Petrie polygons was later extended to semiregular polytopes.
The regular duals, {p,q} and {q,p}, are contained within the same projected Petrie polygon. In the images of dual compounds on the right it can be seen that their Petrie polygons have rectangular intersections in the points where the edges touch the common midsphere.
Square  Hexagon  Decagon  

tetrahedron {3,3}  cube {4,3}  octahedron {3,4}  dodecahedron {5,3}  icosahedron {3,5} 
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. 
The Petrie polygons of the Kepler–Poinsot polyhedra are hexagons {6} and decagrams {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 the order7 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.
{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)  {3,3,5} 600cell 30 sides V:(30,30,30,30,0)  {5,3,3} 120cell 30 sides V:((30,60)^{3},60^{3},30,60,0) 
The Petrie polygon projections are useful for the visualization of polytopes of dimension four and higher.
A hypercube of dimension n has a Petrie polygon of size 2n, which is also the number of its facets.
So each of the (n − 1)cubes forming its surface has n − 1 sides of the Petrie polygon among its edges.
Hypercubes  

The 1cubes's Petrie digon looks identical to the 1cube. But the 1cube has a single edge, while the digon has two. The images show how the Petrie polygon for dimension n + 1 can be constructed from that for dimension n:
(For n = 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 
This table represents Petrie polygon projections of 3 regular families (simplex, hypercube, orthoplex), and the exceptional Lie group E_{n} which generate semiregular and uniform polytopes for dimensions 4 to 8.
Table of irreducible polytope families  

Family n  nsimplex  nhypercube  northoplex  ndemicube  1_{k2}  2_{k1}  k_{21}  pentagonal polytope  
Group  A_{n}  B_{n} 

 H_{n}  
2    pgon (example: p=7)  Hexagon  Pentagon  
3  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron  Icosahedron  
4  5cell   16cell   24cell  120cell  600cell  
5  5simplex  5cube  5orthoplex  5demicube  
6  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
7  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
8  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
9  9simplex  9cube  9orthoplex  9demicube  
10  10simplex  10cube  10orthoplex  10demicube 
In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
In geometry, a 4polytope is a fourdimensional polytope. It is a connected and closed figure, composed of lowerdimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.
In geometry, a tesseract is the fourdimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4polytopes.
In geometry, a hypercube is an ndimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to .
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from Latin stella, "star". Stellation is the reciprocal or dual process to faceting.
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In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or jfaces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n.
In geometry, a crosspolytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ndimensional Euclidean space. A 2dimensional crosspolytope is a square, a 3dimensional crosspolytope is a regular octahedron, and a 4dimensional crosspolytope is a 16cell. Its facets are simplexes of the previous dimension, while the crosspolytope's vertex figure is another crosspolytope from the previous dimension.
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In geometry, the great icosahedron is one of four KeplerPoinsot polyhedra, with Schläfli symbol {3,5⁄2} and CoxeterDynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The interior surface of such a polygon is not uniquely defined.
Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.
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In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.
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