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The Evolution of Coxeter-Dynkin Diagrams

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Polytopes: Abstract, Convex and Computational

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

Abstract

‘Dynkin diagrams’ or ‘Coxeter-Dynkin diagrams’ are graphs whose vertices and edges represent generators and relations (respectively) for so-called Coxeter groups. The generators have period two, and each of the relations specifies the period of the product of two generators. For instance, the group of automorphisms of the 27 lines on the general cubic surface (the Wey1 group E 6) is generated by six involutions, each of which interchanges the two rows of a ‘double six’. An early precursor of the appropriate diagram (see page 4) was used in 1904 by C. Rodenberg to indicate how six double-sixes are related. The same diagrams arise in various connections; for instance, the vertices may represent the mirrors of a kaleidoscope while the edges indicate the angles between pairs of mirrors.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 1A1, Canada

    H. S. M. Coxeter

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  1. H. S. M. Coxeter
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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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Coxeter, H.S.M. (1994). The Evolution of Coxeter-Dynkin Diagrams. 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_2

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  • DOI: https://doi.org/10.1007/978-94-011-0924-6_2

  • 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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