# Why is a circle in a plane surrounded by 6 other circles?

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 $6\times60^\circ=360^\circ$ 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 $$66$$ 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 $$66$$ appears here because the hexagonal lattice has $$66$$-fold symmetry. So a natural question might be whether one can find lattices in two dimensions with, say $$77$$-fold or $$88$$-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: $$66$$-fold symmetry is the best you can do! This is a consequence of the crystallographic restriction theorem. The generalization of the theorem to $$nn$$ dimensions says this: it is possible for a lattice to have $$dd$$-fold symmetry only if $$ϕ(d)≤n\phi(d) \le n$$, where $$ϕ\phi$$ is Euler’s totient function.

The generalization implies that you still cannot do better than $$66$$-fold symmetry in $$33$$ dimensions. There are two natural lattice packings in $$33$$ dimensions, which both occur in molecules and crystals in nature and which both have $$66$$-fold symmetry, and it turns out that these are the densest sphere packings in $$33$$ dimensions. It also turns out that they give the correct kissing number in $$33$$ dimensions, which is $$1212$$ (see the wiki article).

In $$44$$ dimensions, the kissing number is $$2424$$, and I believe the corresponding packing is a lattice packing coming from a lattice with $$88$$-fold symmetry, which is possible in $$44$$ dimensions. In higher dimensions, only two other kissing numbers are known: $$88$$ dimensions, where the $$E8E_8$$ lattice gives kissing number $$240240$$, and $$2424$$ dimensions, where the Leech lattice gives kissing number $$196560196560$$! 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.