Fred lunnon wrote:
Update to search results:
Oh right! I left running my dumb search for 5 points in dim 3 including a pair with unit separation. Today it reached the nice round stopping number of 100 (maximum length of an edge indicent to the unit-separation pair). Time to kill that search, though -- it's now taking about 8 hours for each increment of that radius. That's what O(n^5) gets ya. For the record, here are the 18 instances it found: 1 [1,24,24,23,23,46,16,16,32,30] 2 [1,25,25,25,25,42,2,2,24,24] 3 [1,40,40,31,31,12,26,26,42,40] 4 [1,41,41,41,41,54,31,31,64,70] 5 [1,49,49,46,46,54,46,46,54,86] 6 [1,68,68,58,58,36,32,32,39,45] 7 [1,70,70,12,12,71,8,8,68,5] 8 [1,74,74,52,52,84,52,52,105,27] 9 [1,74,74,74,74,147,52,52,84,105] 10 [1,75,75,59,59,21,55,55,120,109] 11 [1,78,78,28,28,85,10,10,71,36] 12 [1,79,79,79,79,134,6,6,76,76] 13 [1,85,85,40,40,93,23,23,96,21] 14 [1,85,85,85,85,96,13,13,96,96] 15 [1,85,85,85,85,156,13,13,84,96] 16 [1,86,86,40,40,84,40,40,125,57] 17 [1,90,90,38,38,56,10,10,84,36] 18 [1,95,95,70,70,47,37,37,96,89] All of these are, of necessity, "isosceles" in that they have the mirror symmetry that exchanges the two points at unit distance; three of them (2, 5, 14) are doubly isosceles, having a second plane of mirror symmetry as well. Number 14 seem particularly cool: two points have separation 1, and the other three form an equilateral triangle with edge length 96. I wonder if there's an infinite family of similar examples? And Fred, congratulations on finding
[36, 33, 30, 27, 18, 15, 21, 30, 30, 27, 21, 18, 27, 18, 13] You should make a model of it!
--Michael Kleber ps: it occurred to me the other day that every Euler brick leads to a set of seven integer-separation points in three dimensions -- the origin and six more arranged as the vertices of an octahedron. But of course this isn't proper. We could drop the origin and get six points with no three collinar, but the requirement that no four be coplanar shoots this out of the water. (Surely Fred Helenius realized this as soon as the discussion started, but some of us are a little slow...) -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.