On 12/12/06, Michael Reid <reid@math.ucf.edu> wrote:
i also do not know about dan's follow-up question:
| And what about the higher-dimensional analogues: | | Q3: What is the maximum number of points in R^n, no hyperplane | (i.e., n-1 dimensional plane) containing n+1 of them, such that all | interpoint distances are integers?
Well, I had a little look at flat rational pentatopes, or sets of 5 points in Euclidean 3-space with integer distances. They seem much easier to find than planar charts --- in particular, a dumb search program came up immediately with the following examples [AB, AC, BC, AD, BD, CD, AE, BE, CE, DE] = [3, 3, 3, 2, 2, 2, 2, 2, 2, 2] [4, 3, 2, 4, 2, 3, 2, 3, 2, 4] [4, 3, 3, 4, 2, 3, 2, 3, 3, 4] [4, 4, 4, 4, 2, 3, 2, 4, 3, 3] [4, 4, 4, 4, 4, 2, 2, 4, 3, 4] [for which independent verification would be appreciated!] All except the first are conspheric. Rejecting the 120 possible symmetries is a technical problem yet to be completely sorted; till then a longer list is not very practicable. Fred Lunnon