Mathematics.

Covering the plane with congruent circles and leaving no gaps forces the circles to overlap. The thinnest such covering — the arrangement with the least overlap, achieved by centering circles on a hexagonal lattice — has density 2π/√27 ≈ 1.209, meaning the circles' total area exceeds the plane's area by about 21%.

The sphere packing problem asks what fraction of space can be filled by non-overlapping equal spheres. In eight dimensions, the extraordinarily symmetric E8 lattice achieves the provably optimal arrangement — a landmark 2016 result.

The kissing number problem asks how many equal, non-overlapping spheres can simultaneously touch a central sphere of the same size — a question that gets dramatically harder as dimension increases, and is only exactly solved in a handful of dimensions.

Ramsey's theorem guarantees that large enough structures always contain order: however you two-color the edges of a large enough complete graph, a monochromatic triangle becomes unavoidable. R(3,3) is the smallest such threshold.

The exact two-color Ramsey number R(3,4) is 9.

The exact two-color Ramsey number R(3,5) is 14.

The exact two-color Ramsey number R(3,6) is 18.

The exact two-color Ramsey number R(3,7) is 23.

The exact two-color Ramsey number R(3,8) is 28.

The exact two-color Ramsey number R(3,9) is 36.

The exact two-color Ramsey number R(4,4) is 18.

The exact two-color Ramsey number R(4,5) is 25.

The current verified interval for the open Ramsey number R(4,6) is 36–40.

The current verified interval for the open Ramsey number R(5,5) is 43–46.

The exact three-color triangle Ramsey number is 17.

The Leech lattice packing is optimal among all congruent-sphere packings in R^24.

The exact Euclidean kissing number in dimension 1 is 2.

The exact Euclidean kissing number in dimension 2 is 6.

The exact Euclidean kissing number in dimension 3 is 12.

The exact Euclidean kissing number in dimension 8 is 240.

The exact Euclidean kissing number in dimension 24 is 196560.

The hexagonal packing is the densest congruent-circle packing in the Euclidean plane.