Touching simplices

  • Martin Aigner
  • Günter M. Ziegler

Abstract

This is an old and very natural question. We shall call f(d) the answer to this problem, and record f (1) = 2, which is trivial. For d = 2 the configuration of four triangles in the margin shows f (2) ≥ 4. There is no similar configuration with five triangles, because from this the dual graph construction, which for our example with four triangles yields a planar drawing of K 4, would give a planar embedding of K 5, which is impossible (see page 67). Thus we have
$$f(2) = 4$$
.

Keywords

Dual Graph Graph Construction Transversal Line Similar Configuration Zero Entry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    F. Bagemihl: A conjecture concerning neighboring tetrahedra, Amer. Math. Monthly 63 (1956) 328–329.MathSciNetMATHCrossRefGoogle Scholar
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    V. J. D. Baston: Some Properties of Polyhedra in Euclidean Space, Perga-mon Press, Oxford 1965.MATHGoogle Scholar
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    M. A. Perles: At most 2 d+1 neighborly simplices in E d, Annals of Discrete Math. 20 (1984), 253–254.MathSciNetGoogle Scholar
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    J. Zaks: Neighborly families of2 d d-simplices in E d, Geometriae Dedicata 11 (1981), 279–296.MathSciNetCrossRefGoogle Scholar
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    J. Zaks: No Nine Neighborly Tetrahedra Exist, Memoirs Amer. Math. Soc. No. 447, Vol. 91, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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