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 61). Thus we have
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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
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© 2001 Springer-Verlag Berlin Heidelberg
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Aigner, M., Ziegler, G.M. (2001). Touching simplices. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04315-8_12
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