"A Fine Mess," by Dana Mackenzie. NewScientist, 25 May 2002, pages 32-35.
This article discusses some new research into finding the densest spherepacking methods in hyperbolic space. The article warms up to the topic bystarting with an easier question: What is the densest packing of circles in thea regular, flat plane (which mathematicians call the Euclidean plane)? Usingcoins of equal size on a table top, one can get a good feel for this problemand its solution, namely, that the most efficient packing puts each circle incontact with exactly six neighboring circles. In this way, the circles fitinto a neat hexagonal tiling of the plane. Things get a lot more wild andwoolly in the hyperbolic plane, in which the ratio of a circle's circumferenceto its diameter depends on the circle's size; by contrast, in the Euclideanplane, the ratio is always pi. It turns out that, unlike the tidy hexagonalpacking in the plane, the most efficient packings in hyperbolic space are usually irregular. It isalso harder to formulate the notion of ``densest packing'' in hyperbolic space.
--- Allyn Jackson