On 7/22/2011 6:42 PM, Fred lunnon wrote:
I withdraw my earlier ill-considered riposte. [Memo to self: start brain before engaging mouth / pen / keyboard.]
WFL Fred:
As far as I am concerned, the considerable mileage I have gotten from your numerous insightful posts grants you standing absolution for the rare faux pas. I was pretty sure the tesseract mapped to the grid, but I wasn't smart enough to confirm it. So rather than assume it was true and reveal what an idiot I am, I tossed the question out for expert analysis, and you ended up taking the bullet for me. My apologies. So, I suppose a motivation is in order. I had the following conjecture, probably nothing more than a corollary to projective geometers: Given an n-dimensional hypercube H, it is possible to orthogonally project H to some suitably coordinated complex plane P so that some vertex V of H maps to 0 and the vertices H-edge-adjacent to V map onto the n-th roots of unity. I leave it to the geometers to state this correctly, and if true, I imagine it is a fairly elementary theorem of projective geometry. Now I could see the conjecture was true in dimensions 0 through 3, but I wanted a warm fuzzy about dimension 4 before I was willing to accept it was true for higher dimensions. The dimension 4 conjecture required the tesseract vertices and edges to map to the aforementioned 2x2 grid, which I guess is confirmed. So suppose the conjecture is indeed true. Then the vertices of H map to the sums of subsets of the nth roots of unity, which implies that the number of vertices of H that map to 0 is Sloane's A103314(n).