Petrie polygon

In geometry, a Petrie polygon is a skew polygon such that every two consecutive sides (but no three) belong to a face of a regular polyhedron.

This definition extends to higher regular polytopes. A Petrie polygon for an "n"-polytope is a skew polygon such that every "(n-1)" consecutive sides (but no "n") belong to a facet of a regular polytope.

The construction of a Petrie polygon is done via an orthogonal projection onto a plane in such a way that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. These polygons and projected graphs are useful in visualizing symmetric structure of the higher dimensional regular polytopes.

A Petrie polygon of a regular polygon {"p"} trivially has "p" sides as itself.

History

John Flinders Petrie was the only son of Sir W. M. Flinders Petrie, the great Egyptologist. 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 realized the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. He was a lifelong friend of Coxeter, who named these polygons after him.

The idea of Petrie polygons was later extended to semiregular polytopes.

In 1972, a few months after his retirement, Petrie was killed by a car while attempting to cross a motorway near his home in Surrey.

The Petrie polygons of the regular polyhedra

The Petrie polygon of the regular polyhedron {"p", "q"} has "h" sides, where :cos²(π/"h") = cos²(π/"p") + cos²(π/"q")

The regular duals, {p,q} and {q,p}, are contained within the same projected Petrie polygon.

The hypercube and orthoplex families

And the "n"-orthoplex family, {3^"n"−2, 4}, are projected into regular 2"n"-gons with all vertices on the boundary. All vertices are connected by edges except opposite ones.

The semiregular E-polytope family

The semiregular k₂₁ polytopes E5-E8, {3^"n"−3,2,1}, k₂₁

References

* Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
* Coxeter, H. S. M. "The Beauty of Geometry: Twelve Essays" (1999), Dover Publications ISBN 99-35678
* Coxeter, H.S.M.; "Regular complex polytopes" (1974). Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons
* Coxeter, H. S. M. "Petrie Polygons." 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 ")
* Kaleidoscopes: Selected Writings of H. S. M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] (p 31 (24-cell), p 36, p 161 (definition))
* Coxeter, H.S.M.; "Regular complex polytopes" (1974).
* Ball, W. W. R. and Coxeter, H. S. M. "Mathematical Recreations and Essays", 13th ed. New York: Dover, 1987. (p. 135)

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