×

Constructions for regular polytopes. (English) Zbl 0689.51009

Throughout this article, a regular polytope is a regular incidence- polytope as introduced by L. Danzer and the second author in Geom Dedicata 13, 295-308 (1982; Zbl 0505.51019) where the authormophism group A(P) acts transively on the flags of the (abstract) polytope P. In the paper a general method is described how to construct regular polytopes and their groups from other ones. These so-called mixing operations replace the involutions \(\sigma_ 0,...,\sigma_{m-1}\) generating A(P) by other generators \(\rho_ 0,...,\rho_{d-1}\) given in terms of the \(\sigma_ i's.\)
In various cases it is possible to interpret the group generated by \(\rho_ 0,...,\rho_{d-1}\) as the automorphism group of a certain regular polytope. Particular cases are the Petrie operation, the Facetting operation, the Halving operation, the Skewing and the Blending operation. Similar procedures - called twisting operations - on twisted Coxeter groups have been discussed by the authors in Math. Z. 201, 209- 226 (1989; Zbl 0646.51023) and in Adv. Math. (to appear).
Reviewer: W.Kühnel

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
52Bxx Polytopes and polyhedra
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Buekenhout, F., Diagrams for geometries and groups, J. Combin. Theory Ser. A, 27, 121-151 (1979) · Zbl 0419.51003
[2] Colbourn, C. J.; Weiss, A. I., A census of regular 3-polystroma arising from honeycombs, Discrete Math., 50, 29-36 (1984) · Zbl 0539.51014
[3] Coxeter, H. S.M, The abstract groups \(R^m = S^{m\) · Zbl 0014.15103
[4] Coxeter, H. S.M, (Proc. London Math. Soc., 43 (1937)), 33-62, (2) · Zbl 0016.27101
[5] Coxeter, H. S.M, Groups generated by unitary reflections of period two, Canad. J. Math., 9, 243-272 (1957) · Zbl 0077.25101
[6] Coxeter, H. S.M, Regular Polytopes (1973), Dover: Dover New York · Zbl 0258.05119
[7] Coxeter, H. S.M, The 600-cell {3, 3, 5} as a Shadow of \(4_{21}\), the \(E_8\) Polytope (1981), Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 31
[8] Coxeter, H. S.M; Moser, W. O.J, Generators and Relations for Discrete Groups (1980), Springer: Springer Berlin · Zbl 0422.20001
[9] Danzer, L., Regular incidence-complexes and dimensionally unbounded sequences of such (I), Ann. Discrete Math., 20, 115-127 (1984), II in preparation · Zbl 0559.51012
[10] Danzer, L.; Schulte, E., Reguläre Inzidenzkomplexe I, Geom. Dedicata, 13, 295-308 (1982) · Zbl 0505.51019
[11] Dress, A. W.M, A combinatorial theory of Grünbaum’s new regular polyhedra, Part I. Grünbaum’s new regular polyhedra and their automorphism group, Aequationes Math., 23, 252-265 (1981) · Zbl 0506.51010
[12] Dress, A. W.M, A combinatorial theory of Grünbaum’s new regular polyhedra, Part II. Complete enumeration, Aequationes Math., 29, 222-243 (1985) · Zbl 0588.51022
[13] Dress, A. W.M, Regular polytopes and equivariant tessellations from a combinatorial point of view, (Proc. Conf. Algebraic Topology, Göttingen. Proc. Conf. Algebraic Topology, Göttingen, 1984. Proc. Conf. Algebraic Topology, Göttingen. Proc. Conf. Algebraic Topology, Göttingen, 1984, Lecture Notes in Math., Vol. 1172 (1985), Springer-Verlag: Springer-Verlag New York, Berlin), 56-72 · Zbl 0574.51024
[14] Grünbaum, B., Regular polyhedra—old and new, Aequationes Math., 16, 1-20 (1977) · Zbl 0381.51012
[15] Grünbaum, B., Regularity of graphs, complexes and designs, (Problèmes combinatoire et théorie des graphes. Problèmes combinatoire et théorie des graphes, Colloq. Int. CNRS, Vol. 260 (1977), CNRS: CNRS Orsay), 191-197 · Zbl 0419.05017
[16] Jones, G. A.; Thornton, J. S., Operations on maps, and outer automorphisms, J. Combin. Theory Ser. B, 35, 93-103 (1983) · Zbl 0509.57001
[17] McMullen, P., Combinatorially regular polytopes, Mathematika, 14, 142-150 (1967) · Zbl 0155.50002
[18] McMullen, P., Regular star-polytopes and a theorem of Hess, (Proc. London Math. Soc., 18 (1968)), 577-596, (3) · Zbl 0176.19603
[19] McMullen, P., Realizations of regular polytopes, Aequationes Math., 37, 38-56 (1989) · Zbl 0676.51008
[20] McMullen, P.; Schulte, E., Regular polytopes from twisted Coxeter groups, Mathematische Zeitschrift, 201, 209-226 (1989) · Zbl 0646.51023
[21] P. McMullen and E. SchulteAdv. in Math.; P. McMullen and E. SchulteAdv. in Math. · Zbl 0711.51007
[22] McMullen, P.; Schulte, E., Self-dual regular 4-polytopes and their Petrie-Coxeter polyhedra, Results in Math., 12, 366-375 (1987) · Zbl 0649.52005
[23] McMullen, P.; Schulte, E.; Wills, J. M., Infinite series of combinatorially regular polyhedra in three-space, Geom. Dedicata, 26, 299-307 (1988) · Zbl 0643.51020
[24] Monson, B., A family of uniform polytopes with symmetric shadows, Geom. Dedicata, 23, 355-363 (1987) · Zbl 0622.52007
[25] Schulte, E., Reguläre Inzidenzkomplexe II, Geom. Dedicata, 14, 33-56 (1983) · Zbl 0518.51011
[26] Schulte, E., Amalgamations of regular incidence-polytopes, (Proc. London Math. Soc., 56 (1988)), 303-328, (3) · Zbl 0609.51018
[27] Shephard, G. C.; Todd, J. A., Finite unitary reflection groups, Canad. J. Math., 6, 274-304 (1954) · Zbl 0055.14305
[28] Tits, J., Buildings of spherical type and finite BN-pairs (1974), Springer: Springer Berlin · Zbl 0295.20047
[29] Tits, J., A local approach to buildings, (Davis, Ch; Grünbaum, B.; Sherk, F. A., The Geometric Vein (The Coxeter-Festschrift) (1981), Springer: Springer Berlin), 519-547 · Zbl 0496.51001
[30] Weiss, A. I., Incidence-polytopes of type {6, 3, 3}, Geom. Dedicata, 20, 147-155 (1986) · Zbl 0589.51033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.