14 Apr
2016
14 Apr
'16
1:08 p.m.
Given a regular (or any) tetrahedron in R^3 with its face planes extended indefinitely will divide R^3 into a certain number of compartments. A related situation is that of the 2-sphere S^2, where the right spherical triangle (i.e., having all angles equal to 90º) has its 1-faces extended indefinitely to become great circles. Clearly that will divide S^2 into 2^3 = 8 compartments. Clearly, doing the same with the right spherical n-simplex (i.e., having all dihedral angles equal to 90º) in the n-sphere S^n will divide S^n into 2^n compartments. Puzzle: ------- Given an n-simplex in R^n with its face-hyperplanes extended indefinitely, how many compartments will they divide R^n into ? —Dan