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I was hoping someone could recommend a good introduction to the theory of sphere packing. I know that this is a problem that has received some attention lately, due to the solution of this problem in some higher dimensions. I was just wondering how to get into this literature, so I was hoping to find a couple good books or references that provide some intuition as well as the mathematical approaches to the problem.

I am a grad student in statistics, so I have a fair background in applied mathematics including analysis, measure theory, odes, and pdes, etc.

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  • $\begingroup$ There's the classic SPLAG. Sphere packings, lattices and groups, by Conway & Sloane. It is heavy on the algebra (due to the several connections as well as the authors preferences), and not an easy read without a solid background in algebra. Perhaps a bit more of a comprehensive reference book? $\endgroup$
    – Jyrki Lahtonen
    Commented Dec 23, 2018 at 6:13
  • $\begingroup$ @JyrkiLahtonen thanks for the info. Yeah, my background in algebra is limited to just Dummit and Foote. Would that be enough, or would this book require much more than that? I can take a look though, thanks for the tip. $\endgroup$
    – krishnab
    Commented Dec 23, 2018 at 6:26
  • $\begingroup$ Nice short video on the Leech lattice by Richard Borcherds, posted just two days ago: youtube.com/watch?v=ycpmMnO3-Uk $\endgroup$
    – J W
    Commented Jan 28, 2022 at 13:08
  • $\begingroup$ Worth mentioning a classic "Packing and Covering" by C. A. Rogers (1964) $\endgroup$
    – orangeskid
    Commented Jan 29, 2022 at 4:50
  • $\begingroup$ For a gentle introduction, requiring very little background, see Matthew Jenssen's 2021 talk: youtube.com/watch?v=PlBQTSVQ8sY. (Probably too gentle for the OP, but could be of interest to others.) $\endgroup$
    – J W
    Commented Jan 29, 2022 at 18:16

1 Answer 1

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The literature hints in the comments are already very good. When it comes to Sphere Packings, you can hardly get around Conway's & Sloane's book "Sphere packings, lattices and groups".

The following sources might serve as good complementary material:

"Sphere Packing, Lewis Carroll, and Reversi" by Martin Gardner. It published by Cambridge and AMS (which are known for very good books). It is not that heavy in terms of algebraic definitions and structures, hence it provides a really gentle introduction:

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"Lectures on Sphere Arrangements—the Discrete Geometric Side" by Károly Bezdek. It is more Geometry-orientated and thus a bit easier to read for people new to the topic. Along the fundamentals, Unit Sphere Packings (including proofs) the book also deals with Ball-Polyhedra and Spindle Convex Bodies, with Coverings by Cylinders, and with Conjectures such as the Kneser–Poulsen Conjecture and Alexander’s Conjecture:

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"From Error-Correcting Codes Through Sphere Packings to Simple Groups" by Thomas M. Thompson. This book is a nice introduction to Sphere Packings from Group Theoretical point of view. It deals with Coding Theory, Leech Lattice and Simple Groups:

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"Dense Sphere Packings: A Blueprint for Formal Proofs" by Thomas Hales. This is another good Cambridge book on Sphere Packings. It goes through geometrical topics such as Trigonometry, Volume, Hypermap, Fan, and Packing. It contains supporting visualizations, schematic illustrations and reads comfortably too:

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Hopefully these additional literature hints can serve as additional help to dive into the topic of Sphere Packings.

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