- Petrie polygon
In
geometry , a Petrie polygon is askew polygon such that every two consecutive sides (but no three) belong to a face of aregular polyhedron .This definition extends to higher
regular polytope s. A Petrie polygon for an "n"-polytope is askew 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 aregular 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 ofSir W. M. Flinders Petrie , the greatEgyptologist . 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. andCoxeter , H. S. M. "Mathematical Recreations and Essays", 13th ed. New York: Dover, 1987. (p. 135)External links
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