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I understand almost nothing about sphere packings and kissing numbers, but I am very curious to learn about the Newton - Gregory discussion of the problem. In particular, I wonder if Newton's guess was a "blind guess" or an educated guess with a certain logic behind it. So if someone knows a good article that focuses on the Newton - Gregory discussion please give a link.

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"In 1694, a famous discussion between two of the leading scientists of the day - Isaac Newton and David Gregory - took place on the campus of Cambridge University. Their dispute concerned the "kissing problem."" So starts Newton and the Kissing Problem chapter from Szpiro's book Kepler's Conjecture. Pretty much so start (and quickly end) most descriptions of the incident in modern papers on the kissing problem. Not one of them cites a contemporary source, many cite... Szpiro. One exception, to its credit, is Niederman's Number Freak.

The story is an embellished tell-tale. The "famous discussion" was a passing mention in a conversation about astronomy, there was no "dispute", and we are not even sure if Newton guessed right (but it is curious how everyone who retells the story assumes that he was). I suppose inserting the names of Newton and Gregory adds to the "historical respectability" of the kissing problem. Here is from Casselman's Difficulties of Kissing in Three Dimensions:

"The history of the problem is obscure. It is commonly said that in a discussion that took place in Cambridge the Scottish astronomer and mathe- matician David Gregory asserted that 13 spheres could be placed in contact with a central sphere, while Isaac Newton claimed that only 12 were possible. evidence for exactly what was said in this discussion is murky. The first published reference to it that I know of is in the third volume of Newton’s correspondence, edited by H. W. Turnbull, which came out in 1961. There is an entry for May 4, 1694, one of several Latin memoranda written about that day by Gregory, summarizing a conversation with Newton on the distribution of stars of various magnitudes. On the question of 12 versus 13, the entry does not support what is commonly said. Two distinct possibilities are not mentioned, and the most plausible reading is that Newton himself thought that 13 spheres surrounding a fourteenth was a possibility! More likely, some would think, is that Gregory didn’t understand what Newton had said in an apparently rather rapid discourse.

Turnbull refers to a more elaborate entry in a notebook of Gregory kept at Christ Church, Oxford, but at least one person’s attempt to locate that entry where Turnbull said it should be was unsuccessful. In any event, Turnbull’s paraphrase suggests that there is nothing important there not already mentioned in the published memorandum. Other puzzling features of this story are that the 1953 paper by Schütte and van der Waerden refers to a Newton-Gregory discussion, and in 1956 John Leech referred in more detail to the Christ Church notebook. These both appeared several years before the correspondence of Newton appeared in print. What was the source of their information? To paraphrase a familiar dictum of the Mattel Toy Company, history is hard.

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