Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T21:11:35.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Free Extensions of Chiral Polytopes

Published online by Cambridge University Press:  20 November 2018

Egon Schulte  and
Asia Ivić Weiss
Egon Schulte
Affiliation:
Northeastern University, Boston, Massachusetts 02115, U.S.A.
Asia Ivić Weiss
Affiliation:
York University, North York, Ontario, M3J 1P3
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Abstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

Keywords

51M2052A25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 3 , 01 June 1995 , pp. 641 - 654
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Coxeter, H.S.M., Regular Polytopes, 3rd éd., Dover, New York, 1973.Google Scholar
2. Coxeter, H.S.M., Twisted Honeycombs, Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, Rhode Island, 1970.Google Scholar
3. Coxeter, H.S.M., , Regular honeycombs in hyperbolic space, Twelve Geometric Essays, Southern Illinois University Press, Carbondale, 1968.Google Scholar
4. Coxeter, H.S.M. and Moser, W.O.J., Generators and Relations for Discrete Groups, 4th éd., Springer, Berlin, 1980.Google Scholar
5. Danzer, L. and Schulte, E., Regulàre Inzidenzkomplexe I, Geom. Dedicata 13(1982), 295–208.Google Scholar
6. Garbe, D., Uber die regulâren Zerlegungen geschlossener orientierbarer Flâchen, J. Reine Angew. Math. 237(1969), 39–55.Google Scholar
7. Garbe, D., A remark on nonsymmetric compact Riemann surfaces, Archiv. Math. 30(1978), 435–437.Google Scholar
8. Griinbaum, B., Regularity of graphs, complexes and designs, Problèmes Combinatoire et Théorie des Graphes, Coll. Internat. CNRS 260, Orsay, 1977. 191–197.Google Scholar
9. Lyndon, R.C. and Schupp, P.E., Combinatorial Group Theory, Springer, Berlin, 1977.Google Scholar
10. Magnus, W., Noneuclidean Tesselations and Their Groups, Academic Press, New York, 1974.Google Scholar
11. Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory, 2nd éd., Dover, New York, 1976.Google Scholar
12. Malcev, A.I., On the faithful representations of infinite groups by matrices, (Russian) Math. Sb. 8(1940), 405–422. English transi.: Amer. Math. Soc. Transi. (2) 45(1965), 1–18.Google Scholar
13. McMullen, P., Combinatorially regular polytopes, Mathematika 14(1967), 142–150.Google Scholar
14. McMullen, P. and Schulte, E., Regular polytopes from twisted Coxeter and unitary reflexion groups, Adv. in Math. 82(1990), 35–87.Google Scholar
15. McMullen, P., Hermitian forms and locally toroidal regular polytopes, Adv. in Math. 82(1990), 88–125.Google Scholar
16. McMullen, P., Higher toroidal regular polytopes, Adv. in Math., to appear.Google Scholar
17. McMullen, P., Quotients of polytopes and C-groups, Discrete Comput. Geom. 11(1994), 453–464.Google Scholar
18. McMullen, P., Finite quotients of infinite universal polytopes, Discrete Comput. Geom., DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 6(1991), (eds. Goodman, J.E., Pollack, R. and Steiger, W.), 231–236.Google Scholar
19. McMullen, P., Abstract Regular Polytopes, manuscript in preparation.Google Scholar
20. Nostrand, B., manuscript in preparation.Google Scholar
21. Schulte, E., On arranging regular incidence-complexes as faces of higher-dimensional ones, European J. Combin. 4(1983), 375–384.Google Scholar
22. Schulte, E., Extensions of regular complexes, Finite Geometries, Lecture Notes in Pure and Appl. Math. 103(1985), 289–305.Google Scholar
23. Schulte, E. and Weiss, A.I., Chiral Polytopes, Applied Geometry and Discrete Mathematics (The “Victor Klee Festschrift“), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4(1991), (eds. Gritzmann, P. and B.Sturmfels), 493–516.Google Scholar
24. Schulte, E., Chirality and projective linear groups, Discrete Math. 131(1994), 221–261.Google Scholar
25. Sherk, F.A., The regular maps on a surface of genus 3, Canad. J. Math. 11(1959), 452–480.Google Scholar
26. Sherk, F.A., A family of regular maps of type, ﹛6,6﹜, Canad. Math. Bull. 5(1962), 13–20.Google Scholar
27. Vince, A., Regular combinatorial maps, J. Combin. Theory Ser. B 35(1983), 256–277.Google Scholar
28. Wehrfritz, B.A., Infinite Linear Groups, Springer-Verlag, New York, 1973.Google Scholar
29. Weiss, A.I., Incidence-polytopes with toroidal cells, Discrete Comput. Geom. 4(1989), 55–73.Google Scholar
30. Wilker, J.B., The quaternion formalism for Mobius groups in four or fewer dimensions, Linear Algebra Appl. 19(1993), 99–136.Google Scholar