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Modern Developments in Regular Polytopes

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Polytopes: Abstract, Convex and Computational

Part of the book series: NATO ASI Series ((ASIC,volume 440))

Abstract

While the aim of these lectures is to give an overview of modern developments in the theory of abstract regular polytopes, it seeems helpful first to place the subject in its historical context. We shall do that briefly here.

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References

  1. Bieberbach L., Über die Bewegungsgruppen der euklidischen Räume: erste Abhandlungen. Math. Ann. 70 (1910), 297–336

    Article  MathSciNet  Google Scholar 

  2. Colbourn C.L. and Weiss A.I., A census of regular 3-polystroma arising from honeycombs. Discrete Math. 50 (1984), 29–36

    Article  MathSciNet  MATH  Google Scholar 

  3. Coxeter H.S.M., Regular skew polyhedra in 3 and 4 dimensions and their topological analogues. Proc. London Math. Soc. (2) 43 (1937), 33–62 (Reprinted with amendments in Twelve Geometric Essays, Southern Illinois Univ. Press (Carbondale, 1968), 76-105.)

    MathSciNet  Google Scholar 

  4. Coxeter H.S.M., Twisted honeycombs. Regional Conf. Ser. in Math., Amer. Math. Soc. (Providence, 1970)

    Google Scholar 

  5. Coxeter H.S.M., Regular Polytopes (3rd edition). Dover New York, 1973)

    Google Scholar 

  6. Coxeter H.S.M. and Moser W.O.J., Generators and Relations for Discrete Groups (4th edition). Springer Berlin, 1980)

    Google Scholar 

  7. Danzer L. and Schulte E., Reguläre Inzidenzkomplexe, I. Geom. Ded. 13 (1982), 295–308

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 (1981), 252–265

    Article  MathSciNet  MATH  Google Scholar 

  9. Dress A.W.M. A combinatorial theory of Grünbaum’s new regular polyhedra Part II Complete enumeration. Aequationes Math. 29 1985, 222–243

    Article  MathSciNet  MATH  Google Scholar 

  10. Du Val P., Homographies, Quaternions and Rotations, Oxford University Press (1964)

    Google Scholar 

  11. Engel P., Geometric Crystallography. Reidel (Dordrecht etc., 1986)

    Google Scholar 

  12. Grünbaum B., Regular polyhedra—old and new. Aequationes Math. 16 (1977), 1–20

    Article  MathSciNet  MATH  Google Scholar 

  13. Grünbaum B., Regularity of graphs, complexes and designs. Colloques CNRS, Problèmes Combinatoires et Theorie des Graphes, CNRS (Orsay, 1976), 191–198

    Google Scholar 

  14. Hess E., Über die regulären Polytope höherer Art. Sitz. Ges. Bef. ges. Natur-wiss. Marburg (1885), 31–57

    Google Scholar 

  15. McMullen P., Combinatorially regular polytopes. Mathematika 14 (1967), 142–150

    Article  MathSciNet  MATH  Google Scholar 

  16. McMullen P., Regular star-polytopes, and a theorem of Hess. Proc. London Math. Soc. (3) 18 (1968), 577–596

    Article  MathSciNet  MATH  Google Scholar 

  17. McMullen P., Realizations of regular polytopes. Aequationes Math. 37 (1989), 38–56

    Article  MathSciNet  MATH  Google Scholar 

  18. McMullen P., Nondiscrete regular honeycombs. Chapter 10 in Quasicrystals,Networks and Molecules of Fivefold Symmetry, ed. I. Hargittai (VCH Publishers, New York etc., 1990), 159–179

    Google Scholar 

  19. McMullen P., Realizations of regular apeirotopes. Aequationes Math. (in press)

    Google Scholar 

  20. McMullen P. and Schulte E., Regular polytopes from twisted Coxeter groups. Math. Z. 201 (1989), 209–226

    Article  MathSciNet  MATH  Google Scholar 

  21. McMullen P. and Schulte E., Constructions for regular polytopes. J. Combi-nat. Theory A 53 (1990), 1–28

    Article  MathSciNet  MATH  Google Scholar 

  22. McMullen P. and Schulte E., Regular polytopes from twisted Coxeter groups and unitary reflexion groups. Advances Math. 82 (1990), 35–87

    Article  MathSciNet  MATH  Google Scholar 

  23. McMullen P. and Schulte E., Hermitian forms and locally toroidal regular polytopes. Advances Math. 82 (1990), 88–125

    Article  MathSciNet  MATH  Google Scholar 

  24. McMullen P. and Schulte E., Abstract Regular Polytopes. (In preparation.)

    Google Scholar 

  25. van Oss S.L., Die regelmässige fierdimensionalen Polytope hoherer Art. Verh. Konink. Akad. Weten. Amsterdam (eerste sectie) 12.1 (1915)

    Google Scholar 

  26. Schläfli L., Theorie der vielfachen Kontinuität. Denkschriften der Schweizer-lichen Naturforschenden Gesellschaft 38 (1901), 1–237

    Google Scholar 

  27. Schulte E., Reguläre Inzidenzkomplexe, II. Geom. Ded. 14 (1983), 33–56

    MathSciNet  MATH  Google Scholar 

  28. Schulte E., Reguläre Inzidenzkomplexe, III. Geom. Ded. 14 (1983), 57–79

    MathSciNet  MATH  Google Scholar 

  29. Schulte E., Classification of locally toroidal regular polytopes. This volume

    Google Scholar 

  30. Schulte E. and Weiss A.I., Chiral polytopes. In Applied Geometry and Discrete Mathematics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 4 (1991), 493–516

    MathSciNet  Google Scholar 

  31. Schulte E. and Weiss A.I., Chirality and projective linear groups. Discrete Math. (to appear)

    Google Scholar 

  32. Stringham W.I., Regular figures in n-dimensional space. Amer. J. Math. 3 (1880), 1–14

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, London, WClE 6BT, England

    Peter McMullen

Authors
  1. Peter McMullen
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada

    T. Bisztriczky

  2. Department of Mathematics, University College, London, UK

    P. McMullen

  3. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

    R. Schneider

  4. Department of Mathematics and Statistics, York University, North York, Ontario, Canada

    A. Ivić Weiss

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© 1994 Springer Science+Business Media Dordrecht

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McMullen, P. (1994). Modern Developments in Regular Polytopes. In: Bisztriczky, T., McMullen, P., Schneider, R., Weiss, A.I. (eds) Polytopes: Abstract, Convex and Computational. NATO ASI Series, vol 440. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0924-6_5

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  • DOI: https://doi.org/10.1007/978-94-011-0924-6_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4398-4

  • Online ISBN: 978-94-011-0924-6

  • eBook Packages: Springer Book Archive

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