Thanks, Veit. I agree that "anti-kissing number" is the opposite of sexy. Did not know the word "saturated" in this context. Maybe n-dimensional (or nD) saturation number is good terminology. As for density, I am very impressed by how soon the density gets teensy. My calculation of the density for the 196560 23-spheres kissing the central 23-sphere in R^24 is roughly 0.0021977474957616015403222560573790967651611948928748253205373884442323921+, or in other words those 196560 tightly kissing sphere obscure barely over 1/500 of the surface of the 23-sphere. (Thanks, Wolfram Alpha online.) I'd appreciate if someone would care to check me on this calculation. —Dan Veit Elser wrote: ----- A configuration of unit spheres, either in Euclidean space or kissing a given unit sphere, is called “saturated” when it is not possible to add another unit sphere without intersecting one of the spheres. Torquato and co-workers have speculated that “random” saturated packings (sequentially select positions from the uniform distribution on the remaining volume) achieve the *densest* possible packing in the limit of high dimension. I do not know if anyone has looked into your variant of saturated packing, where one is interested in the *lowest* density (fewest number, in the kissing case). I doubt “anti-kissing number” will catch on. Maybe “blocking number”? The corresponding property of a graph is the “minimum size of a maximal independent set”. A(3) <= 6. When you place 6 spheres at the vertices of the regular octahedron no additional spheres can be placed since the positions closest to the central sphere are at the 8 octahedron faces. But these cannot be kissing since we know a 14 kissing configuration does not exist. I’ll guess A(3) = 6, because the most symmetrical 5-sphere configuration — the triangular bi-pyramid — does not work (3 spheres can be added at the equator). -Veit
On Jun 1, 2020, at 7:07 PM, 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)?