I have to apologize on two counts. First, SPLAG is by both Conway and Sloane, not just Conway. Secondly, I misinterpreted the second question; as far as I recall (and can see through a quick examination), SPLAG only addresses the first question. For those interested, discussion of the first question starts at page 21 in the third edition, but there's a very interesting discussion starting on page 29 that is also related. -tom On Mon, Jun 1, 2020 at 7:07 PM Tomas Rokicki <rokicki@gmail.com> wrote:
Conway's Sphere Packing, Lattices and Groups has an interesting (and surprising!) discussion of this particular case.
On Mon, Jun 1, 2020 at 6:37 PM ed pegg <ed@mathpuzzle.com> wrote:
I'm going to say 6, octahedral arrangement. If one more sphere could fit, then more could fit, but only 12 unit spheres can fit around a unit sphere. --Ed Pegg Jr On Monday, June 1, 2020, 06:08:05 PM CDT, Dan Asimov < dasimov@earthlink.net> wrote:
The kissing number in n dimensions is the maximum number K(n) of non-overlapping unit spheres that can be placed tangent to the one centered at the origin.
Known kissing numbers:
dimension kissing number —————————————————————————— 1 2
2 6
3 12
4 24
8 240
24 196560
No other kissing numbers are known.
A related problem is to find the "anti-kissing number" in each dimension n: the smallest number A(n) of non-overlapping unit spheres in n-space, all tangent to the unit sphere centered at the origin, so that there's no room for any additional non-overlapping unit spheres tangent to the central one.
It's obvious that A(1) = 2 and easy to show that A(2) = 4.
Puzzle: What is A(3)?
—Dan
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