April 2020 - mathfun@mailman.xmission.com

[math-fun] 99.9% solution of SIR equations
by Brad Klee 16 Apr '20

16 Apr '20

I asked the question: to what extent is the logistic iteration, x_{n+1} = x_n + x_n*(1-x_n)*dt, a solution of the SIR equations, S' = -S*I, I' = S*I - r*I? The answer, apparently, is 99.9% when dt in [1/10,1], as is shown in the following plots: https://0x0.st/iQv6.png , where the red points are from the discrete iteration and the blue curve is from the continuous ODE solution. The particular function from dt -> r can be found by minimizing err = abs[I' - (S*I - r*I)], after eliminating S by S = Exp[-Int(I*dt)], and after accounting for the non-zero asymptote of S. One shape parameter is one shape parameter, whether it is "dt" or "r", doesn't seem to matter very much at all. --Brad

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[math-fun] I factored 196883 in bed with my eyes closed. Sort of.
by Bill Gosper 16 Apr '20

16 Apr '20

I quickly found the 59 because 883 - 3・196 = = 883 - 600 +12 = 5⨉59, but then I spazzed dividing it out, and erroneously concluded that it was an error. Then I slogged thru primes until 47⨉4189. (It's becoming infrequent that I am lucid enough to do this.) Then I cheated slightly by opening my eyes and asking Mathematica if 4189 was prime. False? Damn, that 59 was right after all! Coincidentally, the final factor, 71, is the algebraic degree of Conway's Constant, the asymptotic growth rate of the Look-Say Sequence <https://www.youtube.com/watch?v=ea7lJkEhytA>. Since JHC was a notorious mental factorer, he must have attacked 196883 as soon as it was revealed as the "dimension" of the Monster Group <https://www.youtube.com/watch?v=jsSeoGpiWsw>. (After his stroke, he admitted to Rich that he was down to 4-digit numbers.) —rwg In[1545]:= FiniteGroupData["Monster","Order"] Out[1545]= 808017424794512875886459904961710757005754368000000000 In[1546]:= FI@% Out[1546]= 2⁴⁶ 3²⁰ 5⁹ 7⁶ 11² 13³ 17 19 23 29 31 41 47 59 71

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[math-fun] And they called me "the mapfolding king"
by Fred Lunnon 16 Apr '20

16 Apr '20

Has anyone downloaded and run this, preferably on a Mac? It looks a worthwhile project, but under the present circumstances I'm cautious about risking my precious desktop --- https://www.theguardian.com/technology/2020/apr/15/volunteers-create-worlds… https://foldingathome.org/ Many years ago I engaged in a certain amount of subterfuge in order to run my hand-optimised assembly-coded strip-of-stamps folding enumerator for 120 hours straight on one of the largest mainframes then available (the Manchester Atlas I), an escapade which earned me considerable amount of sleep-deprivation at the time and notoriety subsequently. Guess I had the last laugh --- folding has come in from the cold! WFL

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[math-fun] John Conway
by Eugene Salamin 15 Apr '20

15 Apr '20

AMS announcement here. http://www.ams.org/publicoutreach/in-memory/in-memory?utm_source=Informz&ut… It says John died from COVID-19. -- Gene

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[math-fun] Mathematical games and bad drawings
by James Propp 15 Apr '20

15 Apr '20

Ben Orlin (the author of the books "Math With Bad Drawings" and "Change is the Only Constant") told me he’s working on a book about mathematical games and is looking for play-testers. He may already have enough; I don’t know. If this is your kind of thing, email me, and I’ll relay your contact info to him. Jim Propp

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[math-fun] Draft of April 17 essay
by James Propp 15 Apr '20

15 Apr '20

I've put a draft of my upcoming Mathematical Enchantments essay on the web at http://mathenchant.org/060-draft4.pdf Comments are welcome! Thanks, Jim

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[math-fun] Extremely discontinuous function
by Allan Wechsler 14 Apr '20

14 Apr '20

Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example?

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[math-fun] unique smallest number
by Bernie Cosell 14 Apr '20

14 Apr '20

A half century ago when I was doing TSD rallying many rallies had a kind of "participation prize". you got to pick a number and the person with the lowest *unique* number got the prize. It struck me back then as an interesting mathematical problem -- the optimal strategy for "N" players. I had one idea: which if N was small, one of us would simply become the team captain and be the only person to pick 1 and then split the prize. Clearly not an optimal strategy [since the non-captain players could collaborate to have one of them cancel the captain's 1 and then one of them would win with 2 -- and so on with everyone sorta cheating]. For no good reason I got to thinking about this again recently and I wonder: is there a game theory analysis of the optimal strategy for N players? I'm guessing it might be something like "everyone picks a random number from 1 to N" but I haven't a clue.. /Bernie\ Bernie Cosell bernie(a)fantasyfarm.com -- Too many people; too few sheep --

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[math-fun] cutting up a cube
by Neil Sloane 14 Apr '20

14 Apr '20

Take a unit square and cut along the lines joining any two vertices. This cuts the square into 4 pieces. Now take a unit cube and make plane cuts though any three vertices: how many pieces are produced? (I don't know) Same question for a unit d-dimensional cube, where the cuts are along hyperplanes through any d vertices. Best regards Neil 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(a)gmail.com

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[math-fun] reminisces of John Conway
by Neil Sloane 14 Apr '20

14 Apr '20

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: http://neilsloane.com/doc/pub.html and other people I've worked with here: http://neilsloane.com/doc/coauthor.html 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 many other familiar names there too.)

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