Welcome to the 160th edition of Carnival of Mathematics!
Welcome to the 160th edition of Carnival of Mathematics, meta-hosted by the remarkable aperiodical.com.
Following tradition I will open this post with "Following tradition I will open this post with". Thus:
Following tradition I will open this post with something interesting about the number 160.
It is an 'interprime', in other words the arithmetic mean of two consecutive odd primes (in this case 157 and 163). There is an nice recent conjecture of Thomas Ordowski (see oeis.org/A024675) that these arithmetic means are identical in value to the ceiling of the geometric mean (i.e. square root of product of the two primes, rounded up to the nearest integer). My curiosity aroused, I checked the first million odd primes and confirmed that the same is true of the harmonic mean (reciprocal of arithmetic mean of reciprocals).
Means of primes can be prime, like 3 and 7. But no interprime can be prime because ... of how they are defined! But this is getting self-referential again. Time to open the postbag.
Let's get some very sad news out of the way. Alexander Bogomolny, creator of Cut the Knot, has died reported Peter Rowlett for the Aperiodical. Cut the Knot has been a dominant and pioneering presence in the online popular mathematics world for over 20 years. It is irreplaceable. Maybe it won't have to be replaced; but I'm not sure what this might mean. The follow up comments to Peter's post begin to ask this question... Another valediction submitted to Carnival is Remembering Alexander Bogomolny by Patrick Honner and he links to further blog postings. Read them all.
By coincidence, illustrating Alexander's appeal and influence, a submission from Colin Beveridge at Flying Colours Maths, Two coins, one fair, one biased, addresses a typical Cut The Knot question. To quote from Colin's introductory remarks: "...equally surely (thinks the reader who knows Alexander doesn’t set trivial problems), it can’t be that simple." No better tribute to Alexander than that!
It seems right to follow that with some other important news, more upbeat. Fields Medal Winners Announced announces Katie Steckles, again for the Aperiodical. Yes, the 4-yearly not-anything-like-the-nobel-prizes has come round again, and we all know more now than we did about optimal transport, algebric geometry, arithmetic algebraic geometry and whatever it is that Akshay Venkatesh does, which seems to be pretty much everything else in modern mathematics!
And that wasn't the only chance to win big in mathematics this last month! Christian Lawson-Perfect, via the Aperiodical (I promise it's not all Aperiodical this month!) ran The Big Internet Math-Off. Who won this one? Read Christian's report The Big Internet Math-Off - The End and find out. It wasn't Peter Rowlett, that's for sure, but it got him thinking and Second place in a single-elimination tournament (at Aperiodical — where else?) is a fun and intriguing read.
More on the big prizes of mathematics: the ABC conjecture resurfaced in abc news (featuring one of the Fields medal winners as a matter of fact) from Peter Woit at Not Even Wrong. Read through all the comments: no-one gets higher calibre comments than Peter Woit and there's a whiff of controversy among these.
And P vs NP gets an airing from Bill Gasarch at Computational Complexity in Soliciting answers for THIRD survey about P vs NP. The deadline for responding is October 1st. Give it a go, but be warned, it gets pretty technical. Reports from the previous two surveys are here (entries 13 and 35).
(Those two challenges are babies compared to the n-body problem which was known to not even have a solution before either was born! So the focus has to be, as Ari Rubinsztejn says in Verlet Integration - The n-Body Problem, "dynamical systems, differences between numerical methods, and error analysis" which he promises he "combines all in a way that an undergraduate who's taken ODE's could understand." If you are not in that category have a look anyway, the graphics are striking, and follow the link to Ari's general n-body page which has a 10-body simulation that is mesmerising.
OK, back to things which we hope do have solutions albeit yet to materialise (well, depending on your point of view for ABC!). One day they will be real theorems, so important they have whole books written about them. Like Poncelet's Porism, for example. Oliver Nash has a fine submission Poring over Poncelet which will certainly convince you that this bit of early 19th-century gometry well deserves at least two modern monographs (which can find them on this web site's bibliography, under Monographs).
And another example submitted this month is Noether's symmetry theorem, beautifully explained (at many levels!) in Emmy Noether’s revolutionary theorem explained, from kindergarten to PhD by Colin Hunter with the Perimeter Institute. Again my bibliography features two "Noether's theorem" books. Actually, three: it's a measure of Noether's genius that I can also list a book devoted to one of her cornerstone contributions to modern algebra. Noether-symmetry is 100 years old, by the way; if you happen to be in London on 11th September you can go (but register first, via the weblink) to the IMA-LMS's joint celebration.
Where else to find wonderful accounts of important theorems? Of course, via the Kevin Knudson–Evelyn Lamb collaboration My Favorite Theorem! The most recent (marking MFT's 1-year anniversary: congratulations!) is Ingrid Daubechies's Favorite Theorem, one of Bill Tutte's which she says "many people learn in kindergarten" (like Noether's symmetry theorem then!)
But MFT appears bi-monthly (apparently I can say this without being wrong!) and indeed I got two splendid submissions posted by Evelyn. And Ken Ribet's Favorite Theorem is certainly a giant theorem. Actually, I haven't come across a whole book about it. It's in Book 9 of Euclid but arguably book 9 is about constructing even perfect numbers, since that's what it all leads up to.
On July 16th Colin Beveridge's favourite theorem was the remainder theorem and his favourite corollary was the factor theorem. I'm guessing, of course. But I think that's why he explains so convincingly Why the Factor and Remainder Theorem Work. It's that thing isn't it—to teach something really well you just have to believe, in that moment, that it's the best thing in the world. Necessary and sufficient, almost.
Well, OK, a sufficient condition for begin wrong about good mathematics teaching is to say you know what it is! Michael Tang is taking the right approach when he says about his submission Math and English Teaching—two comparisons, "I donít often see blog posts that try to highlight the similarities between math and English pedagogues. This post starts this conversation and I believe it will be of interest to educators."
Or what about this, opening a submission by Mark Dominus on How to explain infinity to kids: "A professor of mine once said to me that all teaching was a process of lying." So, this is the Lies-to-Children idea, which you can read about here on wiki but I think Mark gets the idea across better and more interestingly in a quarter of the space. Forget infinity, even zero has its conceptual issues but The Empty Set is Nothing to Worry About, Peter Lynch reassures us.
Anyway, Tao is pre-eminently what Arnaldo Rodriguez-Gonzalez in On the Benefits of Being a Dumb Tourist would call a 'real' mathematician but at least, in the present instance, not one "too busy writing proofs no one understands". That last aimed at ABC in particular but Arnaldo is generally making a contrast with poor-relation ("bizarre great-uncle") statistics and probability (hey, try reading Tao on the Law of Large numbers!) Anyway real mathematicians are not his pet hate: "This brief essay uses my hatred of using public transportation to show that being a 'dumb tourist' is sometimes better than being a 'smart' one using a light dose of statistics!" An interesting and well-written investigation.
And for an interesting and badly-illustrated investigation into statistics and probability you can rely on Ben Orlin's mathwithbaddrawings.com. In Why Itís So Hard to Surprise a Mathematician he de-surprisifies the birthday paradox using the effective technique of scaling-up. Ben has a book coming out in September which everybody is looking forward to.
Even these friendly posts, deliberately de-emphasising the mathematical content, probably won't seduce the self-proclaimed "Always terrible at mathematics...". The oft-implied "... because I had better things to be good at," often appears to elicit sympathy, which is puzzling for those in whom it elicits scorn. Alexandre Borovik helps tackle the "Why is mathematical illiteracy socially acceptable?" puzzle on quora, repeated on his blog Mathematics under the Microscope. See if you agree with him; post a reply yourself if you do not—it's a conversation which, as Michael Tang observed, is a valuable way to go forward in education.
And while you are over at quora mathematics you may bump in another of Alexandre's contributions: to a 'My Favourite' question but 'visual proof' this time, rather than 'theorem'. See if you can guess his answer before you look. In case you aren't at quora at all, Alexandre has again helpfully repeated his answer on Mathematics under the Microscope.
Mark Dominus would enrage "Always terrible at mathematics" with his comment "Everyone wants to know about how to test for divisibility by 7!" and I doubt the accompanying post Divisibility by 7 would appease them. Luckily it appeared to please his daughter Katara and I liked it too.
And next time I meet "Always terrible" I will paraphrase another submission of Mark's: "Everyone wants to know how to test binary operations on finite sets for associativity". His Operations that are not quite associative provides an example that is missing from a 1997 conference paper. Actually, I found a later version of the paper (200K pdf) which did have an example but I think his is nicer.
Danesh Forouhari tweets puzzles like this one which, while not exactly fitting Carnival's current blog post spec, are fun to think about. In this case the puzzle has been seen before, posed elegantly as problem 747 on Antonio Gutierrez's wonderful gogeometry.com where it gives an example of harmonic mean to rival my opening observation about interprimes. I thought: Cut The Knot will have that too ... but I couldn't find it. Instead, to my delight, I found the geometric mean: a contribution by Miguel Ochoa Sanchez, one of Alexander Bogomolny's very many correspondents.
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