From Internet Maths Aperiodical

The Internet Maths Aperiodical – How to cheat and make the Russian people think you can build the Great Pyramid using a Hungarian folk dance

Hello again. I’m Christian Perfect and it’s time for another Internet Maths Aperiodical.

I’m suffering from the post-holiday blues. I’ve cast my Rubik’s cube aside, unsolved; I’m fed up with the public’s appalling ignorance of Gauss facts; worst of all, I’ve made no progress on my New Year’s resolution to learn a different sorting algorithm each day.

Maybe the Aperiodical can help. Let’s see.


Here’s a bit of almost-maths: Nature published an article titled “Scientists make the ‘perfect’ foam” about some scientists who did exactly no such thing. They’ve found (and produced in real life) a slightly more efficient, much less elegantly described, bubble structure than the incumbent imperfect foam, but still no proof of its idealness. Maybe “Scientists make foam with possibly some imperfections” was refused by the editor in charge of eye-grabbing titles.

Two sets of people appear to have ignored Gauss recently, which is an error that shouldn’t really be made even once. First on my radar were Vladimir Putin’s supporters, whose vote-fiddling antics were made apparent by the divergence of certain graphs from the expected Gaussian distribution. Opposing Gauss-appreciative Russians did not miss the opportunity to make some very nerdy placards. There is some explanation of the exact nature of the tomfoolery at this English-language blog post.

Those Italian scientists who were worried they’d measured neutrinos going faster than light might have been assuming normal distributions where they shouldn’t have, according to this blog post by another high-energy particle physicist. An update to the post says that maybe that didn’t matter, anyway. Ladies and gentlemen: statistics!


Here’s a page entitled, “Watch me SOLVE a 20x20x20 cube!” It begins with a bit of fun counting-maths about what it means to solve such a big cube, then there’s a slideshow of the (simulated) cube being solved, with some explanatory text.

I hear you asking who the titular “me” is. You’re right, it probably isn’t anyone famous. Well, here’s a video of heavyweight biopic champion of the world Will Smith solving a Rubik’s cube on French TV. Will Smith’s best days are behind him, you say? You want a younger, more famous celeb? Then here’s teenage heartthrob-of-the-moment Justin Bieber solving a Rubik’s cube on Spanish TV. Still too old, you say, and you’d prefer the celeb-on-cube action to take place in God’s Own America? Happy to oblige. Here’s a somewhat younger Justin solving the cube for Radio Disney in the US.

But celebrities present an image of unattainable perfection. Surely no real person could solve the Rubik’s cube? But expectations have been raised! You will be looked down upon if you do not have this trendy skill.

Well, cancel the booking at that expensive Hollywood brain clinic because standup mathematician Matt Parker has the answer: he has made a video explaining how any idiot with a couple of working hands can take advantage of group theory to make it look like they can solve Rubik’s cube.


Next up is a page making the radical claim that the Great Pyramid at Giza was not built by aliens like we all thought we knew, but by simple humans with knowledge of just enough geometry to get by. I genuinely have no idea if anything on his site is correct. I present it with no further comment, so that you may enjoy this self-effacing old man’s fresh insights as much as I did.

Did you know that HTML with CSS is Turing-complete? It is. These days, I just assume that anything at least as complicated as a cooking recipe is Turing-complete and I’m not often wrong.


A certain school of thought holds that some people learn more effectively through certain senses. Some people take in facts best through their ears, the theory goes, and some learn quickest from visual stimuli, while others don’t really understand something until it’s been up their nose.

Anyway, I think I learn things best through dance. These joyous choreographies of various sorting algorithms performed by an assortment of Central-European dance troupes certainly taught me how to smile, and a little bit about the natural order of the numbers one to nine.


I’ll finish with some properly interesting maths links.

An Alan H. Schoen of The Internet, USA runs a website called Geometry Garret containing “A Pot-Pourri of People, Pictures, Places, Penrose Patterns, Polyhedra, Polyominoes, Posters, Posies, and Puzzles.”

Somebody recently posted Morpion Solitaire to Morpion Solitaire is a very interesting pen-and-paper game for which the best possible score is still now known. You can click through the links on the metafilter thread, but I wanted to highlight the browser-based version of the game.

I’ve found a few good papers for my Interesting Esoterica collection recently. I’ll share two here.

One presents a zero-knowledge protocol for playing mental poker. If you don’t know what a zero-knowledge protocol is, give your parents a copy of “how to explain zero-knowledge protocols to your children” by Quisquater and Guillou, and then get them to explain it to you.

The other sets out to answer the question, “How does one optimally use a snowblower to clear a given polygonal region?” You’ll notice that nothing in this edition of the Aperiodical so far has been NP-complete. Can you guess how hard the snowblower problem is?


That’s all for now. The Aperiodical has cheered me up and I hope it’s cheered you up too. Thanks for being interested in maths!

The Internet Maths Aperiodical – dinner and a theorem

Hello. I’m Christian Perfect and here are some more maths links, after an almost geological gap of one month and one day. I told you this wasn’t going to be a periodical!

There’s going to be some sexy maths at the bottom of this issue of the Aperiodical. I’m just letting you know now so you can prepare yourself by getting in an appropriate state of mind. Maybe some of the links on the way down there will help with that.

Mathematicians like counting. We like counting how many of a thing exist and how many ways a thing can be done and how many kinds of finite simple group there are. In much the same spirit, this happy lady enumerates in a very pleasing video the twenty-five ways she knows of wearing a scarf fashionably. Is this all of them? Are they all different modulo rotation and translation? I think these questions require rigorous mathematical analysis, hopefully in the form of more snappily-edited videos featuring smiling ladies.

OK, so it was a bit of a stretch to claim that video about scarves was also about maths. But this video contains both Leibniz and scarves, so you know it’s maths. It’s also really interesting on its own merits: it’s a presentation by someone who knits stories into scarves using Morse code.

What’s the biggest number of chicken nuggets that you can’t order at McDonald’s without wasting any? In the 1980s, the answer was 43.

McDonald’s is more interesting than that though. In the US they make a thing called the McRib, a sandwich whose appearances seem as transitory and unpredictable as those of the Virgin Mary or the English summer. This article at The Awl makes the case that McDonald’s uses the introduction of the McRib to influence the price of pork. With a graph! So it’s maths!

One final fluff link before we get into the real maths. Here’s a logic quiz that you are guaranteed to fail because of your own PERFIDIOUS BRAIN.


OK, enough weaksauce links and burgers! Time for real maths!

Sphere-packing is a little bit easier now (and here’s the arXiv paper for those who, like me, get nothing out of pop-science writeups and their awful “hey-I-can-relate-to-that-but-you-haven’t-really-made-the-story-any-clearer” way of analogising abstract results). That isn’t the real maths though: it’s imprecise fiddling by a physicist. Real mathematicians only recognise one kind of hard: NP-hard. And circle-packing is hard.

It’s December now, so you probably want to put up some Christmas decorations. Make mathematical ones! Math Craft, which has been aggregating some really good stuff on how to make mathsy objects since it opened recently, has got you covered. They’ve got instructions for six-sided paper snowflakes, an origami christmas tree and they link to a fun article about the fractal patterns you can find in Christmas tree baubles. They’ve got loads of other interesting stuff as well, and they seem to be posting pretty often, so do have a look around. If you’re thinking about making the Christmas tree, be warned: I got roped into an attempt to make it before a seminar on Thursday and it turned out to be way more effort than I thought it would be. I gave up and I feel no shame about that.

A while ago, the people who make episodes of Futurama made an episode of Futurama where everybody’s minds get swapped. One of the writers came up with a group-theoretic justification for the way they resolve the head-swapping. It was pretty fun, and generated quite a few popular maths pieces in the media. Dana Ernst has written a really fantastic set of slides to go with a talk about the episode and its maths. He uses all the trendy web technologies – embedded videos, MathJax, and so on. He even includes an interactive Sage input so you can play about with the groups involved yourself. This is the future of maths exposition!

My last bit of real maths is something I found via Jeff Erickson. It’s an essay on the topic of wooden train-track sets, answering the question of which layouts of left- and right-turning pieces construct a closed curve. It’s a piece of mathematics so beautiful that you will want to kiss your monitor. I’m going to print it out and give it to people I meet. I think I love it.

Finally, now we’re in the mood, turn down the lights because here’s the sexy maths I promised. It’s a short story called “this is math”, by Joey Comeau.

The Internet Maths Aperiodical – temporarily a periodical

Hello again, here’s another Internet Maths Aperiodical. Sadly, until I write a third edition, this Aperiodical currently has a publication period of thirteen days, so I’ll be as quick as I can with this one and I’ll make sure to post again either before or after another thirteen days have elapsed. I can only apologise for the temporary regularity of service.

Here are some more math links.


Shakespeare wrote loads of plays and sonnets and things. Or maybe he didn’t! Isn’t it fun being a literary scholar?

No, it’s more fun being a mathematician. So here’s a blog post about a theory that Francis Bacon was the real author of Shakespeare’s œuvre and hid encoded messages attesting to that fact in the Shakespeare folio by encoding letters in binary, and then using two different typefaces to represent 1s and 0s. As a description of historical events it is of course completely wrong, but logic dictates that all but one of the theories about Shakespeare have to be incorrect, so it shouldn’t feel too bad about itself. Anyway, the idea behind the bilateral code was good, and the people with the Bacon obsession played an important part in the mathematisation of cryptanalysis at the start of the 20th century.


Moving onwards, I have another stupid code for you, but this one’s so stupid it took some really clever people to crack it. The Copiale Cipher is an 18th-century manuscript which had evaded comprehension until some dudes with a load of computers and some fresh new ideas about how to use them had a go at it. Using some computer analysis the team first showed that the manuscript contained a real language and not nonsense (which is a very interesting field of study in itself) and then found it could be deciphered using simple frequency analysis and automatic clustering. They’ve written up their methods and thinking in a very accessible paper. It turned out that the book was written by an esoteric society with a predilection for hazing rituals. HOW TOTALLY UNEXPECTED.


I listened to a lovely little programme about rabbits on Radio 4 this week. It’s available on the iPlayer. There’s, apparently, a breed of rabbits called the Old English Spot. So the programme was all about the trials and tribulations of the people trying to breed the perfect English Spot, and the trials and tribulations of the people who have to live with the people trying to breed the perfect English Spot.

And it is hard to breed a perfect English, because a perfect English is one which looks exactly like the rabbit captured in the painting on this page. In particular, the spots down the rabbit’s side must be in exact correspondence with the picture.

What’s this got to do with maths? Good old Alan Turing worked out how spotty patterns come to be decades ago. Spots are the product of an ever-changing dynamical system known as reaction-diffusion, and are much more congenital than hereditary. So there will probably never be a perfect English, unless someone with a very good knowledge of reaction-diffusion systems and a steady hand with a pipette gets their hands on one in utero.


I saw this story on the BBC News site about how summer babies have it tough throughout their entire lives. I didn’t care, I was born in January. What I did care about was the following pair of sentences:

This reflects that these August children can be almost a year younger than their September-born classmates.

This age gap has not been closed by the time youngsters are ready to leave secondary schools.


Abbott and Costello redux. Sadly, the text has been changed to “This achievement gap…” since I first looked at it. The point about August children being nearly a year younger than their September-born classmates is still pretty much tautological but really I just wanted to post that Abott and Costello clip.


Finally, here’s something interesting. A while ago Twitter was abuzz with the startling revelation that the decimal representations of 1/7, 2/7, etc. all contained the same digits, cyclically permuted. A chap called Lawrence Brenton has written an article in the College Mathematics Journal explaining what’s going on. I like it a lot. It’s all to do with some simple group theory, which he explains very clearly. He makes a persuasive case that all teachers should know a bit of group theory because it leads to more convincing explanations of this kind of thing than number theory alone could.

So if anybody ever asks you what group theory is good for, tell them about this. It won’t take long.

The Internet Maths Aperiodical – "It’s not class war, it’s math"

Hello. My name is Christian and I spend all day looking at maths links on the web.

At least, that’s what Samuel thinks, so he’s asked me to post some of those links here. This will happen to no fixed schedule, so I’ve decided to call these posts “The Internet Maths Aperiodical.”


Let’s start with an FAQ which surely can’t have been asked frequently enough to merit the title.

“Why are there happy puppies on the cover of this Bayesian textbook?”

“The happy puppies are named Prior, Likelihood, and Posterior. Notice that the Posterior puppy has half-up ears, a compromise between the perky ears of the Prior puppy and the floppy ears of the Likelihood puppy. (The puppy on the back cover is named Evidence. MCMC methods make it unnecessary to explicitly compute the evidence, so that puppy gets sleepy with nothing much to do.) If the puppies bother you, see a solution at this blog entry.”

I can’t believe anyone would disapprove of happy puppies on the cover of a Bayesian textbook. Or any book.


Systems, networks and strategies” (via Metafilter)

A maths course for arts students at the San Francisco Art Institute, which is now about halfway through. Because dynamical systems are very easy to visualise, it looks like a well-chosen syllabus for arts students. The wiki page I linked to is a fun stream-of-consciousness collection of links to fun stuff on the internet related to each topic in the syllabus.


Kill Math

“The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.”

Summary: Engineer with a gift for graphic design has trouble with algebra; says he was only shown symbolic methods in school/college; creates graphical tools to help get a feel for where solutions to systems of equations lie; writes essay about that; gives it an enormously provocative title. Take from that what you will. I don’t think anybody will dispute that sketching is an important way of getting a grip on a maths problem. Given that his evidence mostly consists of dynamical systems, maybe he should have taken the course at SFAI that I linked to above, and chilled out a bit.


Jewish Problems

In Soviet Russia, maths problem solves you! (Feel free to delete this one, Sam, it’s in appallingly bad taste)

In order to get into Moscow State University to read maths, applicants needed to pass an oral exam. Apparently, the examiners had a collection of “impossible to solve” questions they would give only to undesirables, in particular Jews. The questions had very simple answers but were worded in such a way as to make them very hard to solve. This paper by the amicablehappy, and not-at-all-odious Tanya Khovanova, along with Alexey Radul, gives both the problems and their solutions.


Baron Munchausen Redeems Himself: Bounds for a Coin-Weighing Puzzle

While looking at that paper I found this one, also by Tanya Khovanova, describing a coin-weighing puzzle. I think these are fairly well-known now, but the Munchausen framing caught my eye because, as is well known, my favourite number is a Munchausen number. I was planning on featuring a paper from my Interesting Esoterica collection each time I make one of these posts, so this might as well be the first!


Benford’s Law and the Decreasing Reliability of Accounting Data for US Firms

Like skateboarding and ska music, every so often the grand wheel of fashion swings around and Benford’s Law enters the public eye again. This is one of those times, and this article claims that accounting data for corporations have been deviating from Benford’s Law more and more since the 1960s. There’s a lot of humming and hahing in the comments about the applicability of Benford’s Law from both people who understand Benford’s Law and people who clearly don’t, which will either excite or frustrate you, depending on your personal policies towards other people’s wrong opinions. A rather good explanation of the idea behind Benford’s Law was posted in the metafilter thread about this analysis.


Cool Rationals

The Champernowne Constant is a rarity among constants – easy to define, hard to remember the name of. This paper, blogged about by the Math Tourist aka Ivars Peterson, gives a representation of the Champernowne constant derived, as far as I can tell, from a boozy night out. The picture of the author at the end of the paper certainly lends weight to that theory.


And finally, here’s your math masquerading as class war as promised, courtesy of a Mr Obama of the United States.