The Aperiodical

David and I sat down again and talked about maths a bit more. I’m calling this number 1 because it suits both our counting systems: David can call this the first podcast of a new series, and I can say the one we put out under All Squared was number 0. Everyone wins! Here follows a long rambly list of things we talked about, and some things we alluded to too. The button to actually play the podcast is down at the bottom of the post. Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory The probabilistic method Swiss cheeses, rational approximations and universal plane curves (The one with the excellent bibliography) A cheaper Swiss cheese Alice in Switzerland: The life and mathematics of Alice Roth Meromorphic function Carrots for dessert Orange peels and Fresnel integrals Cake Cutting Mechanisms The University of Auckland has lots of kiwis in its logo. Newcastle University only has one lion. Computer analysis of Sprouts with nimbers Nimbers On Numbers and Games How to eat 4/9 of a pizza On the Cookie Monster Problem 100 Essential Things You Didn’t Know You Didn’t Know Garfield’s proof of the Pythagorean theorem Napoleon’s theorem Arithmetic derivative David really does have a big tattoo of $\pi$ on his chest. Tukey tallying There exist infinitely many twin primes iff there are infinitely many primes $p$ such that $(p^2)^{\prime\prime\prime} = 1$. The Princeton Companion to Mathematics (warning: auto-playing “podcast”) A CBE is not quite as worthy as a Knight or a Dame The book with the pictures of nudey ladies is Groupes Stables, by Bruno Poizat. The French edition with the pictures is very hard to get hold of (we had to do an inter-library loan through the university), but the foreword to the English translation is superb, and basically boils down to “je ne regrette rien”. The proof that $\sqrt[n+2]{2}$ is irrational because of Fermat’s Last Theorem, which was retold at MathsJam by Julia Collins. That came from the MathOverflow question, “Awfully sophisticated proof for simple facts”. Congruent number Matrix determinant fact Determinants and Matrices by A.C. Aitken (possibly shonky PDF copy) Pfaffian The 15 stupid proofs that the primes are infinite were published in the latest issue of Paradox (PDF), the magazine of the Melbourne University maths and stats society. They’re on page 17.

We haven’t done one of these for absolutely ages. Since all three of us were at the big MathsJam conference a couple of weekends ago, we decided to introduce a local minimum into the fun curve by sitting down and talking about how this site’s doing. Actually, we ended up talking about the MathsJam baking competition for absolutely ages. When we got round to talking about the site, we mentioned: Council orders maths & Sudoku to be removed from mathematician’s gravestone Prime gaps update All Squared, number 10: maths journalism From the mailbag: dual inversal numbers Carnival of Mathematics 104 Submit a thing to the next Carnival of Mathematics

Evelyn Lamb is a professional mathematician who has taken up journalism on the side. She received the AAAS Mass Media Fellowship last year, and spent the summer writing for the magazine Scientific American. We talked to her about maths journalism, the challenges involved in making advances accessible to a wider audience, and the differences between blogging and print journalism. Here are some links to go with the things we talked about: AAAS Mass Media Fellowship Evelyn Lamb. Evelyn on Twitter. Roots of Unity, Evelyn’s blog at Scientific American. The AMS Blog on Maths Blogs, edited by Evelyn Lamb and Brie Finegold. Solved? 80-year-old puzzle of the infinite sphere, New Scientist‘s coverage of last year’s invariant subspace ‘proof’. Adam Goucher’s blog, Complex Projective 4-Space. Evelyn’s Carnival of Maths post.

We have an unusual All Squared podcast for you this time. My good friend David Cushing has been asking to do a podcast for absolutely ages. We couldn’t decide on a single topic to talk about, so instead I suggested we just sit down and chat about maths in general, like we do when there isn’t a microphone in front of us. We talked for about an hour and a half, but because I’m completely stupid we lost a big chunk of it when the microphone switched off. To make things even worse, we recorded in a room with a ridiculously loud fan, so there’s that to contend with. Anyway, we talked about some fun stuff, so I think it’s worth listening to. Here are some links relevant to the things we talked about. David would like you to know that $5 \times 16017 = 80085$. The book David brought was Topology, by James Munkres. Brouwer’s fixed-point theorem says that for any continuous function $f$ with certain properties mapping a compact convex set into itself there is a point $x_0$ such that $f(x_0) = x_0$. The pancake theorem is referred to by MathWorld as “a two-dimensional version of the ham sandwich theorem”, so CP wins. The ham sandwich theorem says that the volumes of any $n$ $n$-dimensional solids can be simultaneously bisected by an $(n-1)$-dimensional hyperplane. The hairy ball theorem says that there is no nonvanishing continuous tangent vector field on even-dimensional $n$-spheres – there’s always a point on the sphere where the function is zero. The black hole information paradox says that it’s possible for a black hole to destroy information. In 2004 Stephen Hawking conceded his bet that information is destroyed, so CP wins again. (Guess who’s writing this summary) The book CP brought was Only Problems, Not Solutions by Florentin Smarandache. It turns out he’s a bit of a character! The family of sequences which contains a sequence for each digit, except inexplicably 1, was “Primes with $n$ consecutive digits beginning with the digit $D$” Problem 102 of Only Problems… contained this cool diagram: We can’t remember what “the Russian book” was. Sorry! The powerful numbers are sequence A060355 in the OEIS. Paul Erdős made a conjecture on arithmetic progressions. The Bee Gees consisted of brothers Barry, Robin, and Maurice Gibb. That’s three people: a powerful triple. $x^2 – 8y^2 = 1$ is a Pell equation, and the reason why the continued fraction representation of $\sqrt{8}$ generates consecutive pairs of powerful numbers. The Muddy Children puzzle is a good introduction to public announcement logic. The slides we were looking at were “The Muddy Children: a logic for public announcement”, by Jesse Hughes. Analysis vs Algebra predicts eating corn? David was playing Wuzzit Trouble by InnerTube Games. It was reviewed here by Colin Beveridge last month.

This is the second and final part of our interview with Colm Mulcahy. Last week we talked about card magic; in this part we moved on to the subject of Martin Gardner and the gatherings of interesting people associated with his name. We’ve tacked on some blather we recorded about the British Science Festival in Newcastle to the end of this podcast. Listen in to hear what we think about maths! (We’re broadly in favour of it.) Here are some links to go with the things we talked about: Martin’s autobiography, Undiluted Hocus Pocus, came out last month. Here’s a review in Plus Magazine. Mathematics Awareness Month in 2014 will be on the theme of “Magic, Mystery and Mathematics”, to celebrate Martin Gardner’s centenary. The Gathering 4 Gardner happens every two years. The next one is in 2014, but it’s invitation only! Celebrations of Mind happen all round the world to carry on the Gardnerian spirit. You can look at a map of all the events and register your own at the official site. Colm’s book Mathematical Card Magic: Fifty-Two New Effects is published by CRC Press, priced £19.99/$29.95 and available from the booksellers in general.

Colm Mulcahy is an original Aperiodical contributor (Aperiodicontributor?) and friend of the site. He’s spent the last year and a bit writing his new book, Mathematical Card Magic: Fifty-Two New Effects. It came out a few weeks ago, so we thought it was a good opportunity to talk to him and find out just what’s so great about mathematical magic tricks. Actually, we had that thought quite a while ago and if we’d been the least bit organised this podcast would’ve come out the same day as the book. As it happened, we first arranged to talk to Colm back in May, and then it took literally three months before we actually managed to record the interview. … And then it took us three weeks to edit it up and upload it. Sorry! Because Colm had so much interesting stuff to say, we’ve split the interview into two parts. In this first half we talk about the book and mathematical card magic; in the second part, out next week, we talk about Martin Gardner and the Celebration of Mind. Mathematical Card Magic: Fifty-Two New Effects is published by CRC Press, priced £19.99/$29.95.

This number of the All Squared podcast contains the final third of our interview with the inestimable David Singmaster, and then a bit from CP about his favourite book, “A treatise on practical arithmetic, with book-keeping by single entry“, by William Tinwell. The first part of the interview, and plenty of links to go with it, were in Number 5 of the podcast. Here are some links to the things we referred to in this podcast: The Casa di Galileo. The Internet Archive has scanned in Dickson’s History of the Theory of Numbers (and Volume 2 and Volume 3). David’s notes are written in the SCRIPT markup language. If you know anything about it and have an idea of how to put it on the web, please get in touch. The Internet Archive also has a scanned-in copy of the fifth edition of Christian’s favourite book, A Treatise of Practical Arithmetic (the “of” seems to have changed to “on” for the sixth edition). Tynemouth Market. The MathsJam annual conference is attended by not just David, Katie and CP, but a whole load of people who are really passionate about recreational maths in all its forms. Registration for the 2013 meeting has just opened – you know what to do! Finally, here are some pictures of Christian’s favourite book, so you don’t just have to imagine what he was talking about: We hope you enjoy listening to this two-parter. We certainly enjoyed recording it!

Good maths books are simultaneously plentiful and rare. While there are a few classics almost everyone knows about and has copies of (Gardner, Hardy, etc.), the trade in lesser-known maths books is considerably less well-organised. Very few bookshops have well-stocked maths sections, and insipid pop maths books dominate. Unless you hear about a good maths book through word of mouth, you’ll often only encounter it once it’s ended up in a second-hand bookshop, usually a refugee from an emptied maths department library. But books, more than anything else, are where the beauty of maths really manifests itself. It’s where ideas are presented most clearly, after they’ve had time to percolate through a few more brains. We talked to David Singmaster, professor of maths and metagrobologist, about his favourite maths books. Here are some links to the things we referred to in the podcast, along with some bonus extras: The Slocum Puzzle Collection at Indiana University Luca Pacioli on Wikipedia Scans of the manuscript of De Viribus Quantitatis The Conjuring Arts Research Center’s edition of De Viribus Quantitatis The one pile game (Static Nim) Nim on Wikipedia Nimber on Wikipedia Nim Multiplication by Lenstra (link goes to a big PDF) The game that wasn’t solved was Sprouts, which is analysed using nimbers in Computer analysis of Sprouts with nimbers by Lemoine and Viennot Winning Ways for your Mathematical Plays by Berlekamp, Conway and Guy Melancholia by Albrecht Dürer at the British Museum (it’s also the icon for our Arty Maths section over there on the right!) The Blind Abbess and her Nuns at The Puzzle Museum The vertical-horizontal illusion at Wikipedia George Hart’s page on De Divina Proportione (and one on Leonardo da Vinci’s illustrations) Scans of De Divina Proportione at archive.org The drawings CP was thinking of were from the Codex Guelf, and posted at BibliOdyssey. Between 1981 and 1985 David edited and published Cubic Circular. With David’s permission, Jaap Scherphuis has put every issue online. CP recommends Westwood Books in Sedbergh and Barter Books in Alnwick as sources of unusual second-hand maths books. The chap who runs Westwood is an ex-mathematician and does a good job of saving books being thrown out of university libraries. Part 2 will appear next week.

It’s a repeat booking for the Festival of the Spoken Nerd in number 4 (or 16 if you belong to Team All Squared) of our podcast. Standup mathematician Matt Parker joined us to talk about interesting coincidences. Here are some links to the things we referred to in the podcast, along with some bonus extras: The operator precedence problem that makes Katie want to cry Colin Wright has blogged about it Chocolate maths! Robert Munafo’s Notable properties of specific numbers The Online Encyclopedia of Integer Sequences A135650 – Even perfect numbers written in base 2 A002904 – Delete all letters except c,d,i,l,m,v,x from n then read as Roman numeral if possible, otherwise 0 A000012 – The simplest sequence of positive numbers: the all 1’s sequence. The episode of Relatively Prime about interesting number facts, including the OEIS. Matt Parker plugged a couple of things: Festival of the Spoken Nerd Maths Gear Matt’s solo show, The Number Ninja Next time, which should be soon, we’ll be talking to a Fantastic Mystery Guest about our favourite maths books.

Remember, remember, The fourteenth of March. While the previous number of All Squared failed to achieve topicality by appearing several weeks after the event it was about, this time we’ve hit the nail bang on the head with a podcast all about π day… on π day! We chatted to Festival of the Spoken Nerd’s Steve Mould about remembering π – how much can you memorise; how much should you memorise; and if you really insist on memorising it, what’s the best way to do it? Here are some links to the things we referred to in the podcast, along with some bonus extras: The number of this episode is 3, which is exactly π if you don’t think very hard when you read the Bible A chronology of the computation of π Epic π quest sets 10 trillion digit record, at New Scientist Download 4 million digits of π “Using pi calculated out to only 39 decimal places would allow one to compute the circumference of the entire universe to the accuracy of less than the diameter of a hydrogen atom.“ The π Code – π in base 26 ≈ D.DRSQLOL 33,333 digits of π in base-(the 1000 most common words in the English language) Japanese breaks pi memory record on BBC News (in which “Conventionally, 3.14159 is used as pi.”) Japanese man claims new record for memorising ‘pi’ at the Daily Mail (in which “It is usually written out to a maximum of three decimal places, as 3.141, in math textbooks.”) Akira Haraguchi The official world record for memorising π is held by Chao Lu of China, who memorised 67,890 decimal digits. Kolmogorov complexity (Steve had the definition right – it measures just the length of the description of the thing, and doesn’t measure the resources required to interpret that description) Normal number Find your birthday in π Find any string of digits in π Michael Hogg’s method of memorising 100 digits of π Joshua Foer: Moonwalking with Einstein – a talk about a book about memory Irrational sonnets (where the stanzas have 3,1,4,1,5 lines) in French, or in English A mnemonic for π which is also a pangram, in French Previously on The Aperiodical: Random walks on π, a marvellous visualisation of the digits of π. Circular reasoning: who first proved that $C/d$ is a constant? – a historical essay by David Richeson. While Steve was chatting to us, his collaborator Matt Parker was measuring π with pies for Numberphile. Steve Mould has his fingers in, if you’ll excuse one final pun, many pies. Here are some links to some of his projects you might find interesting: Steve’s homepage Festival of the Spoken Nerd Maths Gear Steve’s on Twitter