Is there anything more you can say about your level of confidence in each of these numbers? I can prove V=4 and V=5 pretty easily. With V=6 I got as far as proving that any graph on six vertices with more than 9 edges must contain a triangle, before I decided to write this query. On Mon, Jun 20, 2016 at 10:52 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
In 3D space, with n distinct points, what is the maximal number of unit distance lengths?
I'm trying to build up a sequence for OEIS, but I'm only positive I've got the right values on two or three of the entries. Here's what I have so far.
V -- E -- figure 4 -- 6 -- tetrahedron 5 -- 9 -- triangular bipyramid 6 -- 12 -- octahedron 7 -- 15 -- pentagonal bipyramid 8 -- 18 -- snub disphenoid or Raiskii spindle 9 -- 21 -- triaugmented triangular prism 10 -- 25 -- Nechustan spindle 11 -- 28 -- Augmented Nechustan spindle 12 -- 31 -- Double Pacman spindle 13 -- 36 -- Cuboctahedron + center 14 -- 40 -- Cuboctahedron + center + pyramid 15 -- 45 -- Icosahedron + 3 internal points 16 -- 50 -- Icosahedron + 4 internal points
http://math.stackexchange.com/questions/1830194/maximal-unit-lengths-in-3d-w... has a bit more info.
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