On 12/17/06, Fred lunnon <fred.lunnon@gmail.com> wrote: More thorough investigation of the puzzling phenomenon of large numbers of conspheric rational 3-flat 5-topes reveals the all-too-predictable explanation: an unfinished test in the search program letting through large numbers of improper examples, some of whose tetrahedral cells are planar.
When the dust has settled, only the following (proper) rational examples remain, up to edge 6 --- intriguingly, they are exactly those non-conspheric ones found before!
This raises the question of whether there are conspheric ones with no tetrahedral cells planar or other degeneracies. Presumably there are, so what is the 'smallest' one?
[3, 3, 3, 2, 2, 2, 2, 2, 2, 2], [6, 6, 3, 5, 4, 4, 4, 5, 4, 2], [6, 5, 4, 4, 5, 2, 4, 4, 3, 2], [6, 6, 5, 4, 5, 4, 3, 6, 4, 2], [6, 6, 6, 4, 4, 4, 4, 4, 4, 4], # gcd = 2 [6, 6, 6, 5, 4, 4, 5, 4, 4, 5].
The 3-flat 6-tope discussed earlier has 4 of its 6 5-topes in this list, but the other 2 now turn out to have been improper; so it falls over as well.
Notice that there are (so far) no occurrences of an edge of unit length, in either 2- or 3-space. Is there a theorem here?
Fred Lunnon
Jim Buddenhagen