I was listening to Sphere Packing Solved in Higher Dimensions; A Ukrainian mathematician has solved the centuries-old sphere-packing problem in dimensions eight and 24. and reading the transcript there.
Mathematicians have been studying sphere packings since at least 1611, when Johannes Kepler conjectured that the densest way to pack together equal-sized spheres in space is the familiar pyramidal piling of oranges seen in grocery stores. Despite the problem’s seeming simplicity, it was not settled until 1998, when Thomas Hales, now of the University of Pittsburgh, finally proved Kepler’s conjecture in 250 pages of mathematical arguments combined with mammoth computer calculations.
Higher-dimensional sphere packings are hard to visualize, but they are eminently practical objects: Dense sphere packings are intimately related to the error-correcting codes used by cell phones, space probes and the Internet to send signals through noisy channels. A high-dimensional sphere is easy to define — it’s simply the set of points in the high-dimensional space that are a fixed distance away from a given center point.
The Leech lattice is similarly constructed by adding spheres to a less dense packing, and it was discovered almost as an afterthought. In the 1960s, the British mathematician John Leech was studying a 24-dimensional packing that can be constructed from the “Golay” code, an error-correcting code that was later used to transmit the historic photos of Jupiter and Saturn taken by the Voyager probes. Shortly after Leech’s article about this packing went to press, he noticed that there was room to fit additional spheres into the holes in the packing, and that doing so would double the packing density.
Question: How is stacking oranges in 24 dimensions related to receiving and decoding signals from the Voyagers? It it possible to explain in a relatively simple way to the Space SE community, or find a source that does explain the connection suitable for this site?
Thomas Hales, pictured in 1998, used a computer to prove a famous conjecture about the densest way to stack spheres.