On 7/23/11, David Wilson <davidwwilson@comcast.net> wrote:
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.
Hey, somebody actually reads my posts! The dodgy ones, anyway ...
... 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.
Sticking my neck out again (some people never learn), this looks to be what are called "eutactic stars" in Coxeter's book "Regular Polytopes". I strongly recommend you to take a look at this before spending more time on this topic. There is a eutactic star projecting the 5-cube into a regular decagon, and projecting the 6-cube into an icosahedron. Neither is straightforward to construct, being precursors to Penrose tilings in 2 and 3 dimensions. Fred Lunnon