[seqfan] Some reminiscences of John Conway

Neil Sloane njasloane at gmail.com
Tue Apr 14 07:52:41 CEST 2020

He died on April 11 from the corona virus, and I sent these notes to the
person writing his obituary.

Best regards

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com
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Some comments about working with John Conway.
The most impressive thing was the lack of delay before he responded.  When I would suggest a topic, he would respond right away, and often working together we would solve the problem in a couple of hours.

Of course I suppose this worked because we a lot of common interests in discrete mathematics, number theory, coding theory, group theory, geometry, algebra, probability theory, combinatorics, etc.  Not all mathematics, but a pretty wide swath.

And his swath was probably broader than anyone who ever lived.

As he said once, he knew 1000 times as much mathematics as  -- I honestly forget how that remark ends, but the way I remember it, it was referring to the Princeton Faculty.  Or the people in the Princeton Commons Room at the time.  Or maybe he was telling me about some argument about something.  Anyway, it was true!

I've worked with a lot of people, and he was the fastest at solving a problem, and would pursue a topic as far as it would go.

Which is why we were able to write 50 papers together.
You can see the full list here:

and other people I've worked with here:


But he was the best.

He - or we - were also lucky:  sometimes a problem turns out to be just too hard, but working with him he often took a solution a very long way, further than I believed possible when we started.

Our motto was, go as far as any reasonable man would go, and then a lot further.

One favorite example was the construction of dense lattice packings of spheres by "induction" - or in other words, by building up dense packings by taking the best packing in a certain dimension and stacking layers of such spheres in the densest way to get a packing in the next higher dimension.  So you start on dimension 1, which spheres are matches, and you put them in a row and you have the densest lattice packing in dimension 1.  Easy!  Now use induction!!!

The amazing thing is that we were able to follow this induction out to 48 dimensions.
The paper appeared in the Annals, it is #95 on my publ. list (where it can be downloaded)
 Laminated Lattices, J. H. Conway and N. J. A. Sloane, Annals of Math., 116 (1982), pp. 593-620, A revised version appears as Chapter 6 of ``Sphere Packings, Lattices and Groups'' by J. H. Conway and N. J. A. Sloane, Springer-Verlag, NY, 1988.

We drew a picture of how the packings in dimensions 1,2,3,... are built up, it appeared in that paper, it is also in Chapter 6 of SPLAG, and can be seen here: https://oeis.org/A005135/a005135.pdf in A005135.  I have a better version somewhere, should you ever need it.

We called this picture "The shower", for obvious reasons.  You can see we get a unique lattice packing all the way out to 10 dimensions (which is very surprising), then there are two different but equally dense laminated lattices in 11 dimensions, three in 12 dim., 3 in 13 dim., one of which is, as JHC called it, a mule (no offspring), only 1 in 14-d, 1 in 15-D, ..., and one, the great Leech lattice, in 24-D. 23 in 25 dimensions, and
after that all hell  breaks loose and we could not enumerate them.  But we could follow one path though the shower all the way to 48 dimensions.

What made this possible was what we had found out about the "deep holes" in the Leech lattice.  There are 23 of them, and - another miracle - they are in one-to=one correspondence with the other determinant-one lattices in  24 dimensions, the Niemeier lattices

That's described in these two papers:

# 86
Twenty-Three Constructions for the Leech Lattice, J. H. Conway and N. J. A. Sloane, Proc. Royal Society London, Series A, 381 (1982), pp. 275-283, A revised version appears as Chapter 24 of ``Sphere Packings, Lattices and Groups'' by J. H. Conway and N. J. A. Sloane, Springer-Verlag, NY, 1988.

and #87

The Covering Radius of the Leech Lattice, J. H. Conway, R. A. Parker and N. J. A. Sloane, Proc. Royal Society London, Series A, 380 (1982), pp. 261-290, A revised version appears as Chapter 23 of ``Sphere Packings, Lattices and Groups'' by J. H. Conway and N. J. A. Sloane, Springer-Verlag, NY, 1988.

Susanna reminds me that when we were writing up these results in the Caius College library in Cambridge around 1981, Conway sang "Holey, Holey, Holey ..."

And the names, "shower", the "mule", etc are typical of the fun we had.

That was probably the visit to Cambridge when we wrote five papers, a novel, and a play, except we never finished the novel (or the play - both were jokes!)

John's favorite review of our book Sphere Packings, Lattices and Groups (aka SPLAG) appeared in Advances in Mathematics and said: "This book is the best survey of the best work in one of the best fields of combinatorics written by the best people.  It will make the best reading for the best students interested in the best mathematics that is now going on." 

That's probably enough for now, although there is a lot more.

PS There is a photo of John, Richard Guy and Elwyn Berlekamp at the book launch for Winning Ways on the web site for the e-party we had when the OEIS reached 100K entries - see https://oeis.org/wiki/OEIS_100K_E-Party_(Page_1) - scroll down and click on any of their names. (You can see amny other familiar names there too.)

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