When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What’s the significance of 6? Why not some other numbers?
I’m looking for an answer deeper than “there are 6×60∘=360∘ in a circle, so you can picture it”.
The short answer is “because they don’t work,” but that’s kind of a copout. This is actually quite a deep question. What you’re referring to is sphere packing in two dimensions, specifically the kissing number, and sphere packing is actually quite a sophisticated and active field of mathematical research (in arbitrary dimensions).
Here’s one answer, which isn’t complete but which tells you why 6 is a meaningful number in two dimensions. The packing you refer to is a special type of packing called a lattice packing, which means it comes from an arrangement of regularly spaced points; in this case, the hexagonal lattice. The number 6 appears here because the hexagonal lattice has 6-fold symmetry. So a natural question might be whether one can find lattices in two dimensions with, say 7-fold or 8-fold symmetry, since these might correspond to circle packings with more circles around a given circle. (Intuitively, we expect more symmetric lattices to give rise to denser packings and to packings where each circle has more neighbors.)
The answer is no: 6-fold symmetry is the best you can do! This is a consequence of the crystallographic restriction theorem. The generalization of the theorem to n dimensions says this: it is possible for a lattice to have d-fold symmetry only if ϕ(d)≤n, where ϕ is Euler’s totient function.
The generalization implies that you still cannot do better than 6-fold symmetry in 3 dimensions. There are two natural lattice packings in 3 dimensions, which both occur in molecules and crystals in nature and which both have 6-fold symmetry, and it turns out that these are the densest sphere packings in 3 dimensions. It also turns out that they give the correct kissing number in 3 dimensions, which is 12 (see the wiki article).
In 4 dimensions, the kissing number is 24, and I believe the corresponding packing is a lattice packing coming from a lattice with 8-fold symmetry, which is possible in 4 dimensions. In higher dimensions, only two other kissing numbers are known: 8 dimensions, where the E8 lattice gives kissing number 240, and 24 dimensions, where the Leech lattice gives kissing number 196560! These lattices are really mysterious objects and are related to a host of other mysterious objects in mathematics.
A great reference for this stuff, although it is a little dense, is Conway and Sloane’s Sphere Packing, Lattices, and Groups (Wayback Machine). Edit: And for a very accessible and engaging introduction to symmetry in the plane and in general, I highly recommend Conway, Burgiel, and Goodman-Strauss’s The Symmetries of Things.