I’ll start with some made-up maths. Too often, writers end up putting themselves in a corner and needing a maths problem for their brilliant savant character to solve. There are a couple of considerations when using maths in fiction: it needs to look genuinely difficult, and if you’re making a film it needs to looks good when your genius protagonist scrawls it on a blackboard/napkin/the love interest’s chest. Famously, Good Will Hunting satisfied the latter criterion but not the former – Bill H astounded the professors by solving an aesthetically pleasing but pretty trivial problem from linear algebra. The producers of the recent Sherlock Holmes sequel made quite a hullabaloo of going to the effort to commission a proper period-appropriate bit of maths for Moriarty to use in the bits of the film when everything wasn’t exploding.

Anyway, the reason I brought this up is that Philibert Schogt needed some maths for a book he was writing. In books, unlike in films where you have to show someone furiously scribbling symbols, you can be very vague about what exactly the mathematics involved is. Schogt has uploaded an essay to the arXiv describing his process for making up a plausible bit of non-maths, and how the “wild numbers” he didn’t really describe inspired some mathematicians to make them real.

Ducks! Discrete Mathematics with Ducks! Quack!

If a bird heard the word ‘surd’ from a nerd, would it be disturbed? First we need to know why the nerd purred the word ‘surd’. What if the nerd averred that the word ‘surd’ occurred first, slurred, in Urd(u)? Is that lineage of the word ‘surd’ preferred?

Duh I dunno maybe let’s ask a linguist.

Did you know that as well being an excellent dude, Pafnuty Lvovitch Chebyshev was an excellent dude? It needs saying twice. Back in the day, stepper motors and electrical bitbots hadn’t been invented yet, so if you wanted to draw a shape you had to make a linkage to do it for you. Some proud Russians have taken the time to apply their considerable skills towards making a lovely site about the linkages and mechanisms Chebyshev invented.

Talking of linkages, here’s one that draws the lemniscate of Bernoulli, a.k.a. the infinity sign, for you. Nobody need ever fail to draw the infinity sign properly again.

The famous proof convinces a moron in a hurry that all numbers are interesting. There’s certainly evidence that there are some interesting numbers because people collect them, notably Tanya Khovanova’s Number Gossip and Shamos’ catalogue of the real numbers (warning, huge PDF). They both tend to cover the lower end of the scale though – if a least uninteresting number does exist, it isn’t beyond the bounds of imagination that it’s bigger than anything covered in those fascinating compendia.

Robert Munafo’s collection, Notable Properties of Specific Numbers, can hold its head high. Munafo is a Theodore Roosevelt of number-gathering – everything is fair game, no matter how large or how fearsome. Even the index for skipping between pages is a cheerful tourist of the number line, taking intervals whenever something interesting pops up instead of planning ahead and stopping at evenly-spaced markers. When you get to the last couple of pages, the piles of superscripts necessary to denote numbers make them look like perspective drawings of roads made by a TomTom that’s lost its marbles.

But suppose you’re the worst kind of person and find only one number interesting. I’ve got just the thing for you! Incorrectly ascribing mystical properties to the objects around you becomes as easy as looking at something with this handy golden ratio finder card.

Some people find numbers interesting in a different way. The shapes of the digits themselves are subjects of serious study. I can’t remember why I was looking, but I found a freely available database of samples of handwritten digits provided by Yann LeCun and Corinna Cortes. There isn’t much to see there unless you’re a robot, but I thought it was worth mentioning.

Talking of robots looking at things, here’s something we can all appreciate: the Rijksmuseum recently did a rather good job of redesigning its website, and made all of its data about its collections publicly queryable. Not only have they tagged everything they’ve got, they’ve done it in both Dutch and English! Of course I immediately searched for everything to do with maths. I found these trendy dudes on horseback doing angles, as well as a chap whose efforts to get to sleep counting sheep seem to have got out of hand, and a seventeenth century computer cluster. And that’s just the engravings! I like the Rijksmuseum.

Time for me to lay it out straight, like a tablecloth or a trustworthy ruler. I have in the past, in this very column, expressed my expanding distaste for NP-completeness proofs. I’ve said that there are enough, and that we do not need any more. I was wrong about that, and for being wrong I apologise. Vargomax V. Vargomax’s SIGBOVIK presentation, “Generalised Super Mario Bros. is NP-complete“, is a worthy addition to the academic discourse.

Did you notice at the top of the page that the randy mathematician in Edward Frenkel’s short film was played by the mathematician Edward Frenkel? He seems a pretty cool customer. Another cool mathematician is Allen Knutson, who had sufficient sang-froid to leave the last image from his long-gone office webcam up on his homepage because it shows his hand moving to take it away.

Impossible shapes are pretty nice. Fractals can hold one’s interest. Impossible fractals, as drawn by Cameron Browne, gave my brain a good firm tickling. I think what would really add a lot would be some animations running along them. Maybe someone will do that one day. Actually, constructing fractals is pretty hard, especially if it’s impossible, so it might not be a bad idea to enlist some help from something there are uncountably many of. I certainly don’t feel capable of counting the teeming plenitudes of germbugs with whom we share this existence, so the paper Algorithmic Self-Assembly of DNA Sierpinski Triangles is probably worth a read.

That seems to be the end of the links for now. I’ll end with this plippy-ploppy little video of a reaction-diffusion system trying its hardest to be a big-boy cellular automaton. Ladies and gents, SmoothLife:

Until next time!

That’s right: after literally a third of a year, I’m still Christian Perfect and here’s another Aperiodical Round Up!

I’m going to start with computers and calculators, because here’s a really good one: Thomas Fowler’s ternary calculating machine. It uses balanced ternary arithmetic for a variety of reasons which become very interesting when you build your own calculator. Mark Glusker did build his own calculator; that’s a picture of him on the right, looking quietly satisfied with a job well done. No specimens or drawings of the original calculator exist, so Mr Glusker’s machine is only representative of his idea of how it might have looked.

Ternary could have been the kind of arithmetic all computers used, had things worked out differently for ole Jonny Communist. In the 1950s, Nikolay Brusentsov created a computer based on balanced ternary arithmetic called Setun for Moscow State University. It was a mad paradigms double-whammy: the programming language created for it, DSSP, used Reverse Polish Notation. Some nostalgic/patriotic/quixotic Russians have made a charming online simulator of Setun, complete with all the right flippy-switches and terse, efficient Cyrillic labels. I don’t think anyone will knock on your door if you get it to repeat “DOWN WITH STALIN” in an infinite loop, though; if we’ve learned anything in the last two paragraphs, it’s that the authenticity of historical replicas is by necessity limited.

All this talk of ternary arithmetic and Soviet switch-waggling can confuse a chap. Time was, one could perform any calculation desired of one with a simple tool not mechanically dissimilar to a swanee. Derek’s Virtual Slide Rule Gallery lets you relive that simpler time, with a selection of lovingly recreated instruments which can be slid and ruled to your heart’s content. Derek, I salute you!

Did you know that the card game Magic: The Gathering is Turing-complete? Add it to the long, long list of unexpectedly compute-able objects, somewhere near the bottom: it adds to the requirement for infinite resources a requirement for three infinitely patient friends. This never-ending quest that humanity seems to have taken on to find all the Turing-complete objects is beginning to look less like inventive collation of disparate ideas and more like stamp collecting.

Stamp collecting is a hobby to some, and a disdainful term for uninventive enquiry to others. Why not combine the two? Actually, I’m underselling this. Jeff Miller’s page Images of Mathematicians on Postage Stamps is an enormously extensive collection of probably every stamp featuring the face of a mathematician that has ever been printed. There’s an additional section at the end for stamps on other mathematical topics. Why Turkey ever felt the need to issue a stamp promoting the feat of subtraction might remain a mystery to me until my final day.

The first name in James’s list is Niels Henrik Abel’s. Did you know there’s an awesome statue of Abel in the Palace Park in Oslo? I didn’t before, but now I do. The Math Places blog has its coordinates, as well as those of a few other statues, gravestones and architectural features commemorating sundry other mathematicians.

“Stamp collecting” can refer to any activity involving collecting lots of the same type of thing. Robert Browning Sosman engaged in some scientifico-culinary stamp collecting in the 1930s to create his Gustavademecum, an indulgently geek-amenable restaurant guide “prepared for the convenience of mathematicians, physicists, engineers, and explorers”. Sosman made it his job to visit every restaurant in New York, collecting data on the kind of food served, typical prices, and his opinions on the qualities of the food, the staff and his fellow patrons. Rather than writing up his experiences in pithy essays like a journeyman critic, Sosman did what a scientist does and quantified and tabulated his data, so that the reader might draw his own conclusions using whatever metric he desires. The book glories in its nerdishness: in the “other descriptive data” column, an eccentric selection of symbols, mainly Greek letters, is employed to denote the presence of amenities such as an ice skating rink (Σ), a samba band (σ) or a piano (Mπ). I would love to own a copy.

One of the great feats of mathematical stamp collecting of recent years was the classification of the finite simple groups. The collected proof of the classification is beyond any single human brain, but the results are accessible to anyone. They’re even more accessible now, thanks to this handy Periodic Table of Finite Simple Groups drawn up by Ivan Andrus. It collects all the families of finite simple groups into a rather fetching table, arranged and colour-coded by properties and featuring helpful data such as the mind-bogglingly huge orders of some of the groups.

¡Let’s talk about ! Polyhedra ! now!

Or before we do that, I’ll explain the explosive appearance of the previous sentence: ! Polyhedra ! is a Flickr user who collects lots and lots of picture of origami polyhedra. I was swept along by their enthusiasm for polyhedra and thought I’d join in with some peppy punctuation of my own.

If you feel similarly strongly about polyhedra — so strongly that you’d like to share your appreciation with others — you might like to form a club and hold meetings in your very own icosahedral club house, made from the humblest of construction materials: rolled-up newspapers and a few staples.

A clubhouse is an ideal location to play a family-friendly board game. Might I suggest Spinpossible? Alex and Andrew Sutherland reckon it holds up well to mathematical analysis. If that doesn’t hold your interest, you can pass the time playing MilkSnake, a snake game on non-euclidean surfaces.

Hey, what’s yellow and equivalent to the axiom of choice? Zorn’s lemon. What’s a bit up itself but oddly captivating? ZORNS LEMMA.

ZORNS LEMMA is an experimental film made in 1970 by Hollis Frampton. I’m not sure why it’s called that.

The etymology of Back TUVA Future is a lot easier to discern. It’s a track by Tuvan throat-singing master Ondar, intermingled with clips of superstar scientist Richard Feynman laying down his own sweet raps on the subject of not too much at all.

Thanks to this little gem of a record, Richard Feynman acquired an Erdős-Bacon-Sabbath number of 10.

Maths is quite an old subject. Dudes were thinking up theorems and doodling diagrams well before the invention of things that are today considered indispensable, like clothing that doesn’t have a winding number and responsible library archival policy. Chiefly because of the latter (and possibly because of the former – who knows how many great thoughts had to be thrown away for want of a breast pocket), the thoughts of the Ancients aren’t too well known to today’s student. I’ve found three people who want to fix that.

First, S.G. Dani writing for The Hindu seeks to help us understand Ancient Indian mathematics. Probably less helpfully, Messrs. Arcara and Peters of the Classical Liberal Arts Academy have revived the Quadrivium, so that your child can experience the same program of education that served generations of young scholars so well before the invention of such wicked modern constructs as algebra and place-value notation.

Back when the quadrivium was quotidian, it was natural for the brightest young scholars to be sent up at the end of their preparatory studies to a seat of higher learning. There they would continue their solemn study of the Classics until some outside factor such as marriage, political office or the Plague carried them away.

Today’s Oxonians, freed of the life-sapping chores of rote memorisation and toga-wrapping, seem to have a surfeit of spare time. Some of that time has been reassigned to the Oxford mathematical cake seminar. It does what it says on the tin¹- each week, Oxford maths students take turns to bake cakes. The usual suspects are represented, but one cake really sets itself apart: Alessandro Sisto and Rob Clancy’s trefoil knot complement cake. It’s a feast for the mind.

Finally, I have a puzzle for you. The Magic 11-cell is a Rubik’s cube-a-like which takes place on a four-dimensional polytope. It’ll probably take you a while to solve it, if that’s what you decide to do. I can’t even begin to play with it, thanks to my hue-agnostic eyes, but I trust it’s a real puzzle and not just an arbitrary shape-jumbling mechanism. I can’t be completely sure though.

Until next time!

1. do! you! see! what! I! did‽

It’s certainly been a while since the last Round Up. You might not even have the words to describe just how long it’s been. Maybe the book Naming Infinity will help.

Mathematical truths aren’t only revealed through mystical experiences to weirdo monks in the Russian Steppe — sometimes you can see them with your own eyes! For example, brine shimp need remain a mystery no longer because now Science has allowed you to see the vector fields they produce when flapping their ridiculous little limbs about:

And in case you’re ever trapped in a room with just a sheet of paper, a pair of scissors and some Sellotape, don’t despair! And don’t bother trying to McGyver an escape either; make a model of the $\{7,3\}$-tiling of the hyperbolic plane instead. Because after all, the best way to turn a frown upside down is to move to a space with negative curvature.

Now that you’re comfortable with the hyperbolic plane, solving an infinite 3D hyperbolic version of Rubik’s Cube should be a doddle.

Let’s move from enlightening visualisations of abstract truth to the distinctly less rigorous world of data visualisations. Aldo Cortesi has produced a wonderful selection of visualisations of various computery things using the Hilbert curve. On the right is his Hilbert-curve traversal of Mac OS X’s ksh binary, with values mapped to a Hilbert-curve traversal of the RGB colour cube. He has also made some pretty pictures of sorting algorithms in action and this beautiful depiction of the Hilbert curve itself:

Have you ever wondered how Conway’s Game of Life would look on a torus? Well, wonder no more, for it exists. The answer is rather disappointing – it doesn’t look great. Definitely a candidate for application of the recent isometric 3d embedding of the flat torus.

So much visualisation! Your visual cortex must be close to overheating now. Well, I’m bringing the heat with CourtVision Analytics, “examining the NBA through spatial and visual analytics”, featuring articles such as “Breaking Down Chris Paul” and “The Sad MicroGeography of Rajon Rondo“.

Making origami is a pretty excellent way to live your life. If you’ve decided to pursue that worthy vocation, I have a few links that might help you. “Yes yes, you’ve probably got some patterns for origami polyhedra,” I hear you say, “but I’ve always wanted to fold a surface of revolution and I bet you can’t help me there.” Ever so sorry to disappoint/delight, but there help you I can. I present “ORI-REVO, a Design Tool for 3d Origami of Revolution“. Watch in awe as ply-master Jun Mitani twists three boxes from a single sheet:

At last year’s big MathsJam, Costel Harnasz showed us a way of producing a paper heart with a single cut of a folded piece of paper. I was overjoyed to discover the artform’s Orson Welles, Erik Demaine. He has shown that any pattern of straight-line cuts in a piece of paper can be made by folding and a single complete straight cut. He has written a nice page on the subject with some attractive examples, a little bit of history, and some links to further reading. Following the crease patterns is an intellectual exercise in itself, but worth the effort. I made a butterfly!

Finally on the subject of folded paper, if you’ve got a couple of hours to spare, make a magic ball.

I’ve often wondered if a door-to-door circular slide rule business would have been feasible in the time of the Emperor Hadrian. There are many logistical factors to consider, which is why I was happy to learn of the ORBIS project to create a geospatial network model of the Roman world. Here’s a taster:

The model allows for fourteen different modes of road travel (ox cart, porter, fully loaded mule, foot traveler, army on the march, pack animal with moderate loads, mule cart, camel caravan, rapid military march without baggage, horse with rider on routine travel, routine and accelerated private travel, fast carriage, and horse relay) that generate nine discrete outcomes in terms of speed and three in terms of expense for each road segment.

As well as roads, the Romans have a reputation for laying down the law. Aiming to out-pax the Pax Romana is the MaSC group at UCLA, who are doing some really fascinating stuff with modelling crime in hopes of promoting enlightened policing. They were written up recently in an article in the Sydney Morning Herald with the predictably rubbish title “Formula for laying down the law“.

There are lots of tilings of the plane. Loads and loads. More than a googol, even, which is why it’s good to have a tool to search through them. (You have to have seen what I did there. Did you? I’m very proud.) tilingsearch.org provides exactly that service, including a choose-your-own adventure search method.

Sometimes, I suffer from what I call arithmagnosia – I encounter a number in decimal form that looks familiar but whose symbolic form I can’t quite place. Step forward please, the Inverse Symbolic Calculator. If you give it a number in decimal form, it’ll have a guess at the simplest, closest combination of fractions, surds and constants. Or, you could write them out acrophonically and see if they say anything inspiring.

These days, the science of finding Turing-complete systems has advanced to the point where I’m more often surprised when I find that something isn’t Turing-complete than otherwise. It seems that even your precious bodily fluids aren’t safe: water has jumped on the universal computation bandwagon thanks to Bernard Gitton’s Eaurdinateur.

The next sentence is false. The paradox created by the previous, true sentence will be resolved by one of our operators in time for the next edition of the Aperiodical Round Up. For now, please hold:

“Here” is not where it used to be, so I should explain — The Aperiodical is now the name of a big maths conblogerate, of which these untimely collections of miscellanea occupy a small corner. The first four editions of the Internet Maths Aperiodical are still available on ACMEScience.com, and will be for as long as Samuel wants them there.

So, on with the interesting maths links and so on!

Everybody in every nation likes to wring their hands and complain that their children are falling behind everybody else’s children in maths, because everybody else teaches maths better. I came across a couple of news stories recently in this vein: “Follow Brits and do the maths, says top advisor” in the Sydney Morning Herald, and “Teaching students the British way” in the Deccan Herald. The latter story contains a rather enticing implication that all British children are taught maths using chocolate. Yum.

Bode’s Law is either the most useless mathematical law or the most interesting mathematical mistake in the solar system, according to io9, who changed the title of their article from the former to the latter some time in between the day I bookmarked it and now. Personally, I think Bode’s Law is further evidence for my strongly-held anti-platonist views.

Humanity’s eternal quest to classify its every activity into the sets P, NP-complete and NP-hard continues, with the surely world-changing news that Pacman is NP-hard, and so are some other computer games, and Masyu is NP-complete. Masyu puzzles don’t always have unique solutions, which rules them out of Simon Tatham’s portable puzzle collection and out of the category of real puzzles, in my mind.

I’m now quite sick of NP-completeness proofs. I know they’ve become a sort of tradition in the Aperiodical, but I now find them entirely uninteresting. Will some marvellous new NP result regenerate my enthusiasm before the next instance of this organ? Maybe that question is NP-hard. Goddammit all.

Here are some real real puzzles, though: the Jerry Slocum Mechanical Puzzle Collection is being photographed and archived online. The collection contains over 34,000 puzzles. You’ll have to go to Indiana University to play with them though, which isn’t a constraint Gurmeet’s Delightful Puzzles suffer under. Working your way through Gurmeet’s list, from the easy to the difficult puzzles, might be a satisfying way to spend a portion of your life.

Arts and crafts now, with a proof in the style of Dr Seuss that the halting problem is undecidable by Geoffrey K. Pullum, titled “Scooping the loop snooper.” I didn’t mean to imply that there is a Dr Seuss style of mathematical proof in the previous sentence, by the way: I meant that the poem is funny and rhymes a lot while teaching you things. Hope that clears things up.

I found a post on Google+ about Tchokwé people’s tradition of using algorithmic sand drawings to serve as both mnemonics and illustration for stories. João Figueiredo has provided lots of explanation and links, and a really nice slideshow of pictures.

Do you like maths? Do you like “art”? Are you not entirely sick of seeing the same few popular maths concepts and MC Escher cartoons reproduced in “maths art” pieces? Then watch this short film:

Outside the wider cultural context of every individual item in it having been done to death before Elizabeth ascended to the throne, it’s a pleasant, watchable tour through just about every famous maths thing. Half-heartily recommended!

My final arts item is a gallery of S Harris’s cartoons. S Harris drew the famous “and then a miracle occurs…” cartoon, which is never more than five column inches away from a popular maths article. Don’t associate the cartoons in your mind with the humourless people who keep reprinting that exemplar though, because they’re really good. I can’t link to any of my favourite cartoons individually because, I assume, the cartoonist is worried about copyright infringement. But that’s fine: you won’t regret browsing through the whole lot.

That was a nice arts section. Let’s segue away from it slowly, with some Modern Nomograms. Nomograms are charts which embody particular mathematical formulas, so calculating values just involves plotting a line between the numbers you already know. Anyway, these Modern Nomogram chaps have made up a rather appealing poster to calculate posterior probabilities using Bayes’ Theorem, which I might just buy as a present for the Bayesians in my department.

On her blog, Tanya Khovanova tells us how a theorem becomes a magic trick: just like that!

Interactive proofs form sort of a superset of zero-knowledge proofs because it isn’t necessarily secret knowledge which allows one party to do something the other can’t: superpowers will suffice. A superpower I can appreciate is the ability to distinguish colours. A pleasing, child-accessible (what is it about interactive proofs?) article on Images des mathématiques titled “Je suis daltonien, mais je m’en sors” (“I’m colourblind, but I get by”) recounts a possibly-made-up story of how the author helped his children prove that they can distinguish the colours of his socks better than he could. The article’s in French. Désolé, lecteurs monolingues.

This edition’s interesting esotericum is a breezy paper giving decimal calculations of the continued fractions formed from the sets of primes, double primes, Mersenne primes, etc. The continued fraction whose denominators are the prime numbers is quite coincidentally close to one of Rényi’s parking constants.

Let’s end with a showing of Rubik The Amazing Cube (via metafilter)

You can find those earlier works in their own category at ACME Science.