The result the cranks claimed "We show that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subspace of R^N, the sums of the squares of the original and projected edges are in the ratio N/M." is in fact true provided we randomly rotate the polytope before projecting it, and take expectation values. Now given that that is true, it should also be true for certain subgroups of the full rotation group SO(N). In particular, the symmetry group of the polytope itself (if it is edge transitive) is a reasonable candidate. Now Goucher's counterexample of a Rhombus is not allowed since fails to have edge transitive sym group (at least, if you demand group consist wholy of rotations). To be allowed, rhombus would need to be square. And in that case the claim is true, in fact equivalent to Pythagoras's theorem. I think claim is true for hypercubes more generally (?). An actual counterexample would be the degenerate "polytope" consisting of a single edge, but I guess they'd whine that that was degenerate. Re the larger claim they are cranks, well, see http://www.quantumgravityresearch.org/ it appears to be a weird little research institute founded by a rich wacko, but it is not yet obvious to me they are cranks. In particular I think the claim above Goucher said was "obviously wrong" is likely in fact correct, and with a correct proof, albeit you might want to be a bit more careful about exactly what is meant by "group" (rotations only) and exactly what polytopes are allowed (nondegenerate only). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)