Acute triangulations of polyhedra and the Euclidean space

Authors:
Eryk Kopczynski;Igor Pak;Piotr Przytycki
Affiliations:
Warsaw University, Warsaw, Poland;UCLA, Los Angeles, CA, USA;Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Venue:
Proceedings of the twenty-sixth annual symposium on Computational geometry
Year:
2010

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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 construct nontrivial acute triangulations of all Platonic solids. We also prove nonexistence of an acute triangulation of ℜ4 if all dihedral angles are bounded away from ϒ/2.