Sphere packings, lattices, and groups

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1. heinrichhartmanMay 2020
   Sphere packings are quite a deep and fascinating area of pure mathemaics. It has close relations to coding theory, lattice theory, number theory, modular forms.

   John Conway [1] (who recently passed away due to COVID) wrote a famous book about it: Sphere Packings, Lattices and Groups -- which summarizes the subject concisely.

   A recent (2017) break through, was the proof that the Leech Lattice [2] yields the optimal sphere packing in dimension 24 [3]. The error correcting code constructed from this lattice is the Binary Golay Error Correcting Code [4]. It is able to correct 3 bits for every 24 bits, and maximally efficient in doing so. It was used to communicate with the Voyager probes.

   [1] https://en.wikipedia.org/wiki/John_Horton_Conway [2] https://en.wikipedia.org/wiki/Leech_lattice [3] https://arxiv.org/pdf/1603.06518.pdf [4] https://en.wikipedia.org/wiki/Binary_Golay_code

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2. intuitionistApr 2020
   There were better contemporary mathematicians than John Conway, and there were (a handful of) better contemporary popularizers of mathematics, but no one else came close to doing both nearly as well.

   I met him on a few different occasions when I was very young (middle and high school), and he was always extremely generous with his time, especially with an interlocutor not legally old enough to drink, and clearly brilliant in person. One example of his generosity was his collaboration on a book about triangle centers[0] with the late Steve Sigur, who was not a research mathematician but a high school teacher.

   Of course the Game of Life is an enduring classic, but I’ll also always have fond memories of Winning Ways for Your Mathematical Plays and Sphere Packings, Lattices and Groups (affectionately known as SPLAG). The mathematical world has truly lost a living legend.

   [0] https://books.google.com/books/about/The_Triangle_Book.html?...

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