Acute triangulations of polyhedra and ℝN

Abstract

We study the problem of acute triangulations of convex polyhedra and the space ℝn. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n≥4. Further, we prove that acute triangulations of the space ℝn do not exist for n≥5. In the opposite direction, in ℝ3, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of ℝ4 if all dihedral angles are bounded away from π/2.

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Correspondence to Eryk Kopczyński.

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Partially supported by the University of Minnesota, Université Paul Sabatier and the NSA.

Partially supported by MNiSW grant N201 012 32/0718, MNiSW grant N N201 541738, the Foundation for Polish Science, and ANR grant ZR58.

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Kopczyński, E., Pak, I. & Przytycki, P. Acute triangulations of polyhedra and ℝN . Combinatorica 32, 85–110 (2012). https://doi.org/10.1007/s00493-012-2691-2

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Mathematics Subject Classification (2000)

  • 52B05
  • 52C17
  • 51M20
  • 52B10
  • 52C22