Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)

John Flinders Petrie (1907–1972) was the son of Egyptologists Hilda and 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. Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra:

: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.H.S.M. Coxeter (1937) "Regular skew polyhedral in three and four dimensions and their topological analogues", Proceedings of the London Mathematical Society (2) 43: 33 to 62

In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication.H. S. M. Coxeter, Patrick du Val, H. T. Flather, J. F. Petrie (1938) The Fifty-nine Icosahedra, University of Toronto studies, mathematical series 6: 1–26

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.

{{multiple image

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| image1 = Skeleton pair 4-4, size m, thick.png

| image2 = Skeleton pair 4-4, Petrie, stick, size m.png

| image3 = Skeleton pair 4-4, Petrie, stick, size m, 2-fold square.png

| footer = Two tetrahedra with Petrie squares

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{{multiple image

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| image1 = Skeleton pair 6-8, size m, thick.png

| image2 = Skeleton pair 6-8, Petrie, stick, size m.png

| image3 = Skeleton pair 6-8, Petrie, stick, size m, 3-fold.png

| footer = Cube and octahedron with Petrie hexagons

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{{multiple image

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| image1 = Skeleton pair 12-20, size m.png

| image2 = Skeleton pair 12-20, Petrie, stick, size m.png

| image3 = Skeleton pair 12-20, Petrie, stick, size m, 5-fold.png

| footer = Dodecahedron and icosahedron with Petrie decagons

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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.

{{multiple image

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| image1 = Skeleton pair Gr12 and dual, size s.png

| image2 = Skeleton pair Gr12 and dual, Petrie, stick, size s.png

| image3 = Skeleton pair Gr12 and dual, Petrie, stick, size s, 3-fold.png

| footer = gD and sD with Petrie hexagons

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{{multiple image

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| image1 = Skeleton pair Gr20 and dual, size s.png

| image2 = Skeleton pair Gr20 and dual, Petrie, stick, size s.png

| image3 = Skeleton pair Gr20 and dual, Petrie, stick, size s, 5-fold.png

| footer = gI and gsD with Petrie decagrams

}}

The Petrie polygons of the Kepler–Poinsot polyhedra are hexagons {6} and decagrams {10/3}.

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.

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Infinite regular skew polygons also exist as Petrie polygons of the regular hyperbolic tilings, like the order-7 triangular tiling, {3,7}:

:240px

{{multiple image

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| image1 = Tesseract Schlegel Petrie.png

| image2 = Numbered 4-cube.svg

| footer = The Petrie polygon of the tesseract is an octagon. Every triple of consecutive sides belongs to one of its eight cubic cells.

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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 (the 3-sphere), the Petrie polygon of a regular 4-polytope is a 3-dimensional helix in this surface.

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.

• Petrie dual

{{Polytopes}}

{{reflist}}

• 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 )
• Coxeter, H.S.M. (1974) Regular complex polytopes. Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons
• Ball, W. W. R. and H. S. M. Coxeter (1987) Mathematical Recreations and Essays, 13th ed. New York: Dover. (p. 135)
• Coxeter, H. S. M. (1999) The Beauty of Geometry: Twelve Essays, Dover Publications {{LCCN|99035678}}
• Peter McMullen, Egon Schulte (2002) Abstract Regular Polytopes, Cambridge University Press. {{ISBN|0-521-81496-0}}
• Steinberg, Robert,ON THE NUMBER OF SIDES OF A PETRIE POLYGON, 2018 [https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/on-the-number-of-sides-of-a-petrie-polygon/F66E7EF633FB67A3D0209A5DF9F51195]

{{commons category|Petrie polygons}}

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• {{mathworld | urlname = HypercubeGraph | title = Hypercube graphs}}
• {{mathworld | urlname = CrossPolytope | title = Cross polytope graphs}}
• {{mathworld | urlname = 24-Cell | title = 24-cell graph}}
• {{mathworld | urlname = 120-Cell | title = 120-cell graph}}
• {{mathworld | urlname = 600-Cell | title = 600-cell graph}}
• {{mathworld | urlname = GossetGraph | title = Gosset graph 3_21}}

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Category:Polytopes