Petrie Schemes

Williams, Gordon

Canadian journal of mathematics, Tome 57 (2005) no. 4, pp. 844-870

Résumé

Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grünbaum–Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes.

Détail

DOI : 10.4153/CJM-2005-033-6

Mots-clés : Primary: 52B15, secondary: 52B05, Petrie polygon, polyhedron, polytope, abstract polytope, incidence complex, regular polytope, Coxeter group

Bibliographie

[1] [1] Alexander, J. W., The combinatorial theory of complexes. Ann. of Math. 31(1930), 292–320. Google Scholar

[2] [2] Altshuler, A. and Steinberg, L., Enumeration of the quasisimplicial 3-spheres and 4-polytopes with eight vertices. Pacific J. Math. 113(1984), 269–288. Google Scholar

[3] [3] Barnette, D., Diagrams and Schlegel diagrams. In: Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf.), Gordon and Breach, New York, 1970, pp. 1–4. Google Scholar

[4] [4] Barnette, D., The triangulations of the 3-Sphere with up to 8 vertices. J. Combinatorial Theory Ser. A 14(1973), 37–53. Google Scholar

[5] [5] Bokowski, J., Philippe, C., and Mock, S., On a self dual 3-sphere of Peter McMullen. Period. Math. Hungar. 39(1999), 17–32. Google Scholar

[6] [6] Bokowski, J. and Sturmfels, B., Computational Synthetic Geometry. Lecture Notes in Mathematics 1355, Springer-Verlag, 1989. Google Scholar

[7] [7] Bourbaki, N., Lie Groups and Lie algebras. In: Elements of Mathematics, Ch. 4–6. (Pressley, A., trans.) Springer-Verlag, Berlin, 2002. Google Scholar

[8] [8] Bruggesser, H. and Mani, P., Shellable decompositions of cells and spheres. Math. Scan. 29(1971), 197–205. Google Scholar

[9] [9] Coxeter, H. S. M. Regular skew polyhedra in three and four dimensions, and their topological analogues. Proc. LondonMath. Soc. Series 2 43(1937), 33–62. Google Scholar

[10] [10] Coxeter, H. S. M., Regular Polytopes. Third edition. Dover Publications, New York, 1973. Google Scholar

[11] [11] Coxeter, H. S. M., Twelve Geometric Essays. Southern Illinois University Press, Carbondale, IL, 1968. Google Scholar

[12] [12] Dress, A., A combinatorial theory of Grünbaum's new regular polyhedra, Part I: Grünbaum's new regular polyhedra and their automorphism group. Aequationes Math. 23(1981), 252–265. Google Scholar

[13] [13] Dress, A., A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration. AequationesMath. 29(1985), 222–243. Google Scholar

[14] [14] Danzer, L. and Schulte, E., Reguläre Inzidenzkomplexe I. Geom. Dedicata 13(1982), 295–308. Google Scholar

[15] [15] Grünbaum, B., Convex Polytopes. Interscience Publishers, 1967. Google Scholar

[16] [16] Grünbaum, B., Regular polyhedra—old and new. Aequationes Math. 16(1977), 1–20. Google Scholar

[17] [17] Grünbaum, B. and Sreedharan, V. P., Enumeration of simplicial 4-polytopes with 8 vertices. J. Combinatorial Theory 2(1967), 437–465. Google Scholar

[18] [18] Hartley, M. I., All polytopes are quotients, and isomorphic polytopes are quotients by conjugate subgroups. Discrete Comput. Geom. 21(1999), 289–298. Google Scholar

[19] [19] Humphreys, J. E., Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics 29. Cambridge University Press, Cambridge, 1990. Google Scholar

[20] [20] McMullen, P., On the Combinatorial Structure of Convex Polytopes. Ph.D. dissertation, Univ. Birmingham, 1968. Google Scholar

[21] [21] McMullen, P. and Schulte, E., Constructions for regular polytopes. J. Combin. Theory Ser. A 53(1990), 1–28 . Google Scholar

[22] [22] McMullen, P. and Schulte, E., Higher toroidal polytopes. Adv. Math. 117(1996), 17–51. Google Scholar

[23] [23] McMullen, P. and Schulte, E., Twisted groups and locally toroidal regular polytopes. Trans. Amer.Math. Soc. 348(1996), 1373–1410. Google Scholar

[24] [24] McMullen, P. and Schulte, E., Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications 92. Cambridge University Press, Cambridge, 2002. Google Scholar

[25] [25] Schulte, E., Classification of locally toroidal regular polytopes. In: Polytopes: Abstract, Convex and Computational (Bisztriczky, T., McMullen, P., Schneider, R. and IvicWeiss, A., eds.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer, dordrecht, 1994, pp. 125–154. Google Scholar

[26] [26] Schulte, E. Wolfram Research, Mathematica 5.0.Wolfram Research,Champaign, IL, 2003. Google Scholar

[27] [27] Wilson, S., New Techniques for the Construction of RegularMaps. Ph.D dissertation, University of Washington, 1976. Google Scholar

[28] [28] Ziegler, G M., Lectures on polytopes. Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1991. Google Scholar