# Kissing Number of Spheres in Non-Euclidean Geometry

There has been much work done on the kissing number problem (of determining the greatest number of congruent spheres which can touch a single sphere in a packing) in Euclidean space for dimensions $1$ to $24$ and even some asymptotic work. My question is whether or not the kissing number problem has been studied in Non-Euclidean geometries such as the other seven Thurston Geometries $\mathbb{S}^2 \times \mathbb{R}, \mathbb{H}^2 \times \mathbb{R}, \mathbb{S}^3, \mathbb{H}^3, \widetilde{SL_{2}(\mathbb{R})}$, Nilgeometry, or Solvgeometry.

To understand the cases of spherical or hyperbolic geometry, it is helpful to think in terms of spherical codes. A Euclidean kissing configuration is equivalent to an arrangement of points on a sphere with all angles between them at least 60 degrees (these are the points of tangency with the surrounding spheres). The same construction works in any sphere or hyperbolic space $X$, because the induced geometry on spheres in $X$ is itself spherical geometry. The only thing that varies is the minimal angle, which will generally no longer be 60 degrees. Instead, it will depend on the radius of the kissing spheres, and it can be calculated using the (spherical or hyperbolic) law of cosines.

So the kissing problem in spaces of constant curvature amounts to looking at spherical codes with differing angles. I don't know what happens in the other Thurston geometries. There has been some study of packings in these geometries (see, for example, http://arXiv.org/abs/1210.2202 by Szirmai), but I don't know offhand about optimal kissing numbers.

• @Henry Cohn: I have seen Szirmai's paper regarding packing density in the Thurston Geometries and although it is interesting, does not refer to the kissing number problem. It seems to be unstudied in the more obscure of the Thurston Geometries, so I think I will work on research regarding this topic! The point of my question was to find out if it had been done before, and if not, then I have free range to solve an interesting problem. – Samuel Reid Mar 1 '13 at 16:43
• @Samuel Reid: Szirmai does make one relevant comment, namely that one of the packings he studies in nilgeometry has kissing number $14$. I'd recommend e-mailing him if you would like a more definitive answer than I can give about what's known. This problem does sound worth studying (if you take it up, please let me know your findings). – Henry Cohn Mar 2 '13 at 16:47
• @Henry Cohn: I did not notice the comment about the kissing number in nilgeometry before, thank you for pointing that out! I am already collaborating with Szirmai, so I will ask him about it. It's good to hear that there is interest in this idea, I will send you a message in the future if I end up writing a paper on this topic! – Samuel Reid Mar 2 '13 at 20:51

In one sense, which you are free to see as unimportant, the kissing number in the standard hyperbolic plane is unbounded, as it increases with increasing radii of the disks. One of the Laws of Cosines says $$\cos \alpha = 1 - \frac{1}{1 + \cosh a},$$ where $a$ is the side length of an equilateral triangle and $\alpha$ is the vertex angle. As $a$ increases without bound, $\alpha$ decreases without nonzero lower bound. So, with discs of large enough radius, we can place as many disjoint-interior disks as we like around a given central one. The equilateral triangle has vertices at the centers of three mutually tangent circles.

I'm just saying.

• The point being, the kissing "number" is now replaced with the kissing "function" (of the radius). – Allen Knutson Mar 1 '13 at 11:28