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The Derivation of Schoenberg’s Star-Polytopes from Schoute’s Simplex Nets

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The Geometric Vein

Abstract

On a square billiard table with corners (± 1, ± 1), the path of a ball is easily seen to be periodic if and only if it begins with a line

$$ Xx + Yy = N, $$

where X and Y are integers and |N| < |X| + |Y| [16, p. 82]. Ignoring a trivial case, we shall assume XY ≠ 0. We lose no generality by taking these integers to be positive and relatively prime. After any number of bounces, the path is still of the form

$$ \pm Xx \pm Yy = N \pm 2k, $$

where k is an integer. Among these paths for various values of k, those that come closest to the origin are of the form

$$ \pm Xx \pm Yy = N \pm = N', $$

where 0 ⩽ N′ ⩽ 1. The distance of such a path from the origin is

$$ N'/\sqrt {{X^2} + {Y^2}} . $$

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Author information

Authors and Affiliations

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

    Chandler Davis  & F. A. Sherk  & 

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Branko Grünbaum

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© 1981 Springer-Verlag New York Inc.

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Coxeter, H.S.M. (1981). The Derivation of Schoenberg’s Star-Polytopes from Schoute’s Simplex Nets. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds) The Geometric Vein. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5648-9_9

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  • DOI: https://doi.org/10.1007/978-1-4612-5648-9_9

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-5650-2

  • Online ISBN: 978-1-4612-5648-9

  • eBook Packages: Springer Book Archive

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