2009/09/21 by Eryk Kopczyński, Kopczyński, Eryk, Igor Pak +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG
paper · pdf · doi:10.48550/arxiv.0909.3706
22 pages, 7 figures
arxiv created 2009/11/20 · arxiv updated 2009/12/08
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 pi/2.