Yes — thanks for the correction, Tom: "...will divide S^n into 2^(n+1) compartments" is most definitely what I should have written. —Dan
On Apr 14, 2016, at 1:16 PM, Tom Karzes <karzes@sonic.net> wrote:
Hi Dan,
In the following:
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.
Didn't you mean "...will divide S^n into 2^(n+1) compartments"?
Tom
Dan Asimov writes:
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.
NO! This should be 2^(n+1) compartments, as Tom points out.
Puzzle: ------- Given an n-simplex in R^n with its face-hyperplanes extended indefinitely, how many compartments will they divide R^n into ?
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun