Combinatorica

, Volume 32, Issue 1, pp 85–110 | Cite as

Acute triangulations of polyhedra and ℝN

Original Paper

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.

Mathematics Subject Classification (2000)

52B05 52C17 51M20 52B10 52C22 

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Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2012

Authors and Affiliations

  1. 1.Institute of InformaticsWarsaw UniversityWarsawPoland
  2. 2.Department of MathematicsUCLALos AngelesUSA
  3. 3.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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