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A Hierarchical Classification of Euclidean Polytopes with Regularity Properties

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Polytopes: Abstract, Convex and Computational

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

Abstract

During the last decades, many mathematicians investigated classes of poly topes which are determined by natural weakenings of the definitions of regular polytopes. Examples are the sets of regular-faced polytopes, congruent-faced polytopes and isogonal polytopes. The aim of the present article is to show one possibility of bringing these classes into one hierarchical structure, perhaps stimulating further research with respect to gaps in this structure, natural extensions of its parts or new combinations of the definitions contained. In addition, we give a survey regarding important results on each of these polytope classes. (More related material can be found in a recent survey by the author which, however, does not show this hierarchical structure and the related connections between the polytope classes discussed here, see Martini (1994).)

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  1. Fachbereich Mathematik, Technische Universitaet Chemnitz, Chemnitz, Germany, D-09009

    Horst Martini

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  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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Martini, H. (1994). A Hierarchical Classification of Euclidean Polytopes with Regularity Properties. 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_4

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