Acute triangulations of polyhedra and the Euclidean space
Eryk Kopczyński, Igor Pak, Piotr Przytycki · 2010
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.