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 59). Thus we have
“How many d-dimensional simplices can be positioned in ℝd such that they touch pairwise, that is, such that all their pairwise intersections are (d — 1)-dimensional?”
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References
F. Bagemihl:A conjecture concerning neighboring tetrahedra, Amer. Math. Monthly 63 (1956) 328–329.
V. J. D. Baston: Some Properties of Polyhedra in Euclidean Space, Perga-mon Press, Oxford 1965.
M. A. Perles: At most 2 1+1 neighborly simplices in E d, Annals of Discrete Math. 20 (1984), 253–254.
J. Zaks: Neighborly families of 2` 1 d-simplices in E d, Geometriae Dedicata 11 (1981), 279–296.
J. Zaks: No nine neighborly tetrahedra exist, Memoirs Amer. Math. Soc. 447 (1991).
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Aigner, M., Ziegler, G.M. (1998). Touching simplices. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22343-7_13
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DOI: https://doi.org/10.1007/978-3-662-22343-7_13
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