I don’t think the university maths department I work in has enough art in it. I have gazed covetously upon the walls of other departments I visit, covered with beautiful mathematically-inspired paintings and inspirational posters, serving as a backdrop to cabinets full of geometrical curiosities. I recently suggested to our Head of School that we could buy some art, and he said “That’s a good idea. Send me some suggestions.”

I was pretty delighted with that response, so I spent an enjoyable hour trawling the internet for art that would inspire and enrich our students and staff. We don’t really have anywhere obvious to put sculptures, so I wanted something you can hang on a wall. I had no idea how much money the Head of School was thinking of spending, so I assumed the worst and tried to stick to cheap posters and prints as a starting point. I wasn’t just looking for art – anything to decorate the walls, even if it ends up teaching the students something, is desirable.

My first port of call was my Arty Maths blog. I’ve been collecting nice bits of art that invoke or involve maths (and not art created purely to represent maths) for almost two years now. Unfortunately, it turns out I’ve almost exclusively been collecting sculptures and video works. That meant I had to do some googling!

Because I found some nice things, and in case anyone else is tasked with decorating a maths department and needs ideas, here’s what I found:

Some chaps in America have produced “Modern Nomograms”: charts for quickly doing Bayesian calculations. I’ve always thought they look nice, but I leave it up to the statisticians to judge whether they’re worthwhile from a statistical perspective. The big poster is $25 plus shipping.

Simon C Page is a mathematician turned graphic designer. His geometric prints are very nice. 16″x24” posters are £40.

Brent Yorgey produced these factorisation diagrams a few months ago. They’re quite pretty, they take a while to understand, and he’s shared the code used to make them so we can make posters as big as we like. He mentioned on his blog that he was looking into having proper posters made up, so I’ve emailed him to ask if that ever happened.

Johnny Lin keeps a list of links to mathematical posters on his website. I’ve collected the best ones below.

The Beauty of Mathematics Poster Collection offers a few sets of posters on geometrical topics lying just on the threshold of being interesting. I think the undergrads might like them, but I don’t like the graphic design. £30 for a set of five laminated A2 posters on a single topic.

Peter Jipsen has a few freely-available PDFs of posters on his site. I particularly like the series expansions of $e$ and $\ln 2$. The rest are basically just Beamer slides.

Ivan Andrus has made this fantastic periodic table of the finite simple groups. I think it’s a lovely idea with a decent execution, and the classification of the finite simple groups is the longest proof in history, so I reckon it’s worth commemorating.

I found this company selling prints of decimal representations of the biggest Mersenne primes. They seem to have stopped in 2008; I’m not sure if they just lost interest or realised the primes were getting too big to print at any size. Anyway, I think walking past a ridiculously long prime every day would make people smile. An unframed print is $99 plus shipping.

There’s a site selling rather light-hearted posters of mathematicians. I think quite a few maths departments have copies of these, so we might want to buck the trend, but they’re quite nice designs, if a bit garish. I might email them and ask if they’re planning on doing any more women – Ada Lovelace on her own is a bit of a poor show! Each poster $30.

The Bridges conference is an annual gathering of people working on the connections between maths and art. I had a look through last year’s exhibits, and John Higli and Markus Rissanen stood out. I don’t know how easy it would be to obtain pieces from them, though.

Edmund Harriss does lots of work on connecting art and maths, and could be commissioned to do something if we had more than a poster’s worth of cash. He knows a lot about geometry.

More please!

Now that I’ve compiled this post I can see that I mainly found things which were more educational than aesthetically pleasing. There’s plenty of lovely maths art out there – see the Bridges conference’s enormous collection of exhibits – but it seems the artists either aren’t interested in selling prints, or don’t make it easy to find out that they do. That’s a shame, because there seems to be a big gap in the market between the garish no-design educational posters and beautiful but expensive commissioned pieces.

So, if you’re a mathematical artist who either already sells prints of your work or could be convinced to, please send me an email. Or if you’re not an artist but have a recommendation for something I’ve missed, please post a link in the comments.

But dammit, I’m a mathematician, not a physicist. So, instead of talking about the science of Star Trek yet again, what about the maths of Star Trek? After all, Star Trek is science fiction, but there is no such thing as maths fiction – so any mathematics featured on the show is sure to be on firmer ground. Right? Or as Spock himself says in ‘The Conscience of the King’;

SPOCK: Even in this corner of the galaxy, Captain, two plus two equals four.

Should we even expect much maths to feature on a simple space adventure show? In fact, many interesting mathematical ideas were raised during the show’s short run of 79 episodes, including; the probability we are alone in universe; a paradox that upset 20^th century mathematicians as well as 23^rd century androids; the mathematics of alien and Earth biology; and the most important question of all – when on a dangerous away mission, does the colour of your shirt really affect your chances of survival?

In this article we will focus purely on Star Trek: The Original Series. No movies, no animated series, no Next Generation, no JJ Abrams reboots, and definitely no lens flares.

Kirk does not believe in the no-win scenario

So let’s begin by answering one of the most fundamental mathematical questions about Star Trek: The Original Series:

Being in Starfleet and visiting strange new worlds is dangerous work. The safety of his crew is a top priority for Captain Kirk, especially if it affects the running of his ship.

For example, in ‘The Devil in the Dark’, the crew of the Enterprise are hunting a mysterious creature that has already killed several of the local miners. Kirk, Spock and a team of Enterprise security officers search for the creature in the maze of tunnels under the planet’s surface.

Spock and Kirk inspect the tunnels in ‘The Devil in the Dark’

KIRK: Mister Spock, you are second in command. This will be a dangerous hunt. Either one of us by himself is expendable. Both of us are not.

SPOCK: Captain, there are approximately one hundred of us engaged in this search, against one creature. The odds against you and I both being killed are 2,228.7 to 1.

KIRK: 2,228.7 to 1? Those are pretty good odds, Mister Spock.

SPOCK: And they are of course accurate, Captain.

KIRK: Of course. Well, I hate to use the word, but logically, with those kind of odds, you might as well stay. But please stay out of trouble, Mister Spock.

SPOCK: That is always my intention, Captain.

To work out the odds that both Kirk and Spock die, we can begin by considering what would happen if there were exactly two deaths. Then the probability that this is both Kirk and Spock (in either order) is $2 \left( \frac{1}{100} \times \frac{1}{99} \right) = \frac{1}{4950}$. In the form of odds that’s 4,949 to 1.

This turns out to be a very good approximation. My own calculation showed that if $x$ people die, then the probability that this includes both Kirk and Spock is $^{98}C_{x-2}$ ($98$ choose $x-2$) in $^{100}C_x$ ($100$ choose $x$). If each scenario becomes increasingly less likely by a factor of a half, then the final odds that Kirk and Spock both die works out again to be 4,949 to 1.

Spock’s odds are not quite as favourable and, although Spock doesn’t reveal his methodology, he must be correct because he gives his answer to one decimal place – and that’s despite his not knowing exactly how many men are taking part in the search.

Two of these people are going to die

However, we know Kirk and Spock are never in any real danger – they are the stars of the show after all. And so here we address one of the classic tropes of the original Star Trek. That the red shirt always dies – usually followed by Doctor McCoy’s diagnosis, “he’s dead, Jim!”

But is the trope really true? Is a crewmember in a red shirt (security, engineering and operations) genuinely more likely to die than a blue shirt (science and medical) or a gold shirt (command, helm)?

Analysis of this problem has been tried before, but has been flawed. So let me present my own analysis of the problem.

Looking at the figures, there were a total of 59 crew deaths during the 79 episodes of Star Trek the original series. 8 deaths were of blue shirts, 10 deaths were gold shirts, 16 deaths are unknown crew members, while 25 deaths were red shirts.

So by those figures alone, it appears that red shirts are the ones most at risk. Indeed, ignoring the unknown deaths, the probability of being a red shirt if you die is 58% $\left( \frac{25}{43} \right)$. The odds don’t seem too good – even Joan Collins had a 50-50 chance of survival.

Schrödinger’s girlfriend

But to make any accurate judgement we need to know the total population of red shirts. After all, you expect more red shirts to die if there are more red shirts to begin with. And that turns out to be the case:

We may estimate the probability of dying to be around 13.7% $\left( \frac{59}{430} \right)$. Ignoring the unknown deaths, it appears that 58.1% of deaths are red shirts $\left( \frac{25}{43} \right)$, while only 23.2% of deaths are gold shirts $\left( \frac{10}{43} \right)$.

But there are four times as many red shirts as gold shirts. Of that population, only 10.5% $\left( \frac{25}{239} \right)$ of red shirts died during the three seasons of Star Trek, as opposed to 18.2% $\left( \frac{10}{55} \right)$ of gold shirts!

However, 18 fatalities were security officers, out of a total population of 90, making it the most dangerous department of all. The probability of death if you are a security officer is 20%.

So it is far more dangerous to be a gold shirt than a red shirt – unless you are a security officer, in which case, remember to set phasers on full.

This is the first in a series of three posts about The Maths of Star Trek. The series continues in Part II, which discusses the mathematics of alien biology, and Part III will follow on Friday.

Probably the greatest mathematician of the twentieth century, Paul Erdős … was so eccentric that he made Einstein look normal. He was 11 before he ever tied his shoes, 21 before he ever buttered toast, and died without ever boiling an egg. Erdős lived on the road, traveling from conference to conference, owning nothing but math notebooks and a suitcase or two. His life consisted of math, nothing else.

- Clifford Goldstein, in The Mules That Angels Ride (2005), p. 125

Born in Hungary in 1913, Paul Erdős was the one of most prolific mathematicians of all time: the honour of most pages published goes to Leonhard Euler, but Erdős has the most papers published in collaboration with others – around 1,525. He worked as a nomad, travelling between conferences and other mathematicians’ hospitality, and wherever he went he created fantastic collaborations with the world’s greatest mathematicians.

Another roof, another proof.

- Erdős’ motto, as quoted in A Tribute to Paul Erdős (1990) edited by Alan Baker, Béla Bollobás, A. Hajnal, Preface, p. ix

Erdős worked in many areas of pure mathematics - combinatorics, graph theory, number theory, set theory and probability theory to name a few. He collaborated with 511 different mathematicians, and was always travelling – his possessions fit in a suitcase, and most of his earnings and awards were generally donated to charity.

Erdős knows about more problems than anybody else, and he not only knows about various problems and conjectures, but he also knows the tastes of various mathematicians. So if I get a letter from him giving me three of his conjectures and two of his problems, then it’s sure that these are exactly the kind of conjectures and problems I’m interested in, and these are exactly the kind of questions I may be able to answer.

- Béla Bollobás, of Trinity College, University of Cambridge in N Is a Number: A Portrait of Paul Erdős (1993)

The Aperiodical is of course a pro-everybody enterprise all year round, but it doesn’t hurt to take some time to remind ourselves of the fact that women are just as capable as men of contributing to the field of maths. Incredibly, some people still don’t think this is the case!

So, without further ado, here’s a very quickly assembled list of:

A few women who did or do maths, off the tops of our heads (& Calvin’s)

Hypatia

Geometrix, preserver of the old Greek way of doing things, played by Rachel Weisz in a forgettable movie.

Florence Nightingale

Dataviz pioneer.

Caroline Herschel

Astronomer, first female scientist recognised by the Royal Astronomical Society

Mary Somerville

Translated Laplace, helped bring English maths out of its Newtonian rut.

Emmy Noether

Deserving of a Brad Neely hyperbole song. Intellectual heavyweight, owner of an adjective.

Ada, Countess Lovelace

You can count on this one!

Sophie Germain

You could say she’s… germain to this discussion?

Hazel Perfect

CP’s maths hero, translated Alexandroff’s An Introduction to the Theory of Groups.

Grace Hopper

Rear Admiral, writer of the first programming language compiler and stone-cold coolest woman in the US Navy.

Ingrid Daubechies

Officially (ish) the world’s highest-ranking mathematician.

Kathleen Ollerenshaw

Educational advisor to Thatcher, got her PhD without even writing a thesis, classy Dame (Commander of the Order of the British Empire).

Thousands and thousands of ‘computers’

@BParkEd has tweeted to remind us that “around 80% of the folks at Bletchley Park during WWII were women”. The women at Bletchley Park were mainly not doing any maths though – while many were working as computers, most were collecting and translating messages.

Back in the day, before computers were machines that crunched numbers for us, computers were women that crunched numbers for (almost always) men. What they did wasn’t intellectually trivial, and was rarely acknowledged.

Additionally, women were often employed as amanuenses – many monographs from the early part of the twentieth century begin with a note thanking a female name for her work preparing the author’s handwritten notes for print.

It seems that for a long time, mathematically gifted women were relegated to translating mathematical works from other languages, e.g.:

• Mary Somerville – Laplace into English
• Eleanora Barbapiccola – Descartes into Italian
• Hazel Perfect – Alexandroff into English
• Ada Lovelace – while translating Menebrea’s memoir on Babbage’s analytical engine, wrote ‘notes’ consisting of effectively the first computer programs.

Today

The LMS recently released its report ‘Advancing women in mathematics: good practice in UK university departments’, which we reported on the site. This has statistics on current numbers of women at different stages of the academic career and a discussion of what university departments can do in good practice.

Just under 20% of professors of mathematics in the UK are women, though almost half of all maths lecturers are female.

The next LMS Women in Maths Day will take place 18-19th April. Our page on this quotes:

The event provides an opportunity to meet and talk with women who are active and successful in mathematics. While this is an occasion particularly for women active in mathematics to get together, men are certainly not excluded from this event.

Not one woman has been awarded a Fields medal yet.

Links to further reading

International Women’s Day 2013.

Biographies of Women Mathematicians website.

Women in math project.

Wikipedia’s Category:Women mathematicians.

London Mathematical Society Women in Mathematics page.

Hypatia’s Heritage: A History of Women in Science by Margaret Alic.

Some mathematics, pictured here being hard to illustrate in news coverage

As the heady excitement of the dawn of a forty-eight-Mersenne-prime world dims to a subdued, albeit slightly less factorable, normality, I have taken the opportunity to see what we can learn about the British press’s attitude and ability when it comes to the reporting of big numbers ending in a 1.

Overseas readers may not be aware that the UK’s public service broadcaster, the BBC, is funded by a mandatory annual £145.50 tax on all television-owning households. Therefore, it would be disappointing if some of these funds were not channeled into reporting the discovery in at least five or six separately-produced broadcasts across the organisation’s various radio and television outlets.

I’ll start with BBC radio’s flagship news programme, Today. At 7.20 in the morning on Friday 8th, they ran an interview with the prime’s ‘discoverer’ Curtis Cooper, introduced and conducted by presenter Justin Webb. This was probably the strongest of the $2^2-1$ interviews Cooper conducted for the BBC; Webb asked the most straightforwardly sensible question: what does one actually ask the computer to do to find the primes? Cooper gave a good overview of the maths behind the GIMPS project. The interview benefitted enormously by departing from Today’s house style and not bringing in a second interviewee forced to somehow argue against the large number, in order to inject a pointless and counter-informative facade of balance into proceedings.

Cooper popped up again the previous evening (having finally obtained a prime number large enough to serve as a power source for his time machine) on the BBC World Service’s Newshour. Showing that nobody is immune to doing something mildly stupid when instructed to do so by a person with a clipboard, he acquiesced to read out a portion of the immense integer in an imbecilic made-up “seven-quinnelty-miliibong-quadro-trilli-trilli-thouswart” style, followed by the producer doing the same thing but with less fluent delivery, to the amused chortling of all. Aside from laughter at the general bigness of the number under discussion, Cooper covered most of the same points he was to reiterate the following morning, discussing the use of somewhat smaller primes in RSA encryption and the potential power of similar distributed-computing projects.

Chronic Aperiodical-hassler Matt Parker tweeted on Thursday night that he was due to appear on the World Service around the same time as Cooper – apparently, he did do an interview but it went out on World Service channels not accessible in the UK. Not, as we secretly suspected, that a strange person convinced him their living room was a BBC studio and got him to come in and talk about maths into a cardboard microphone.

A mere two hours later Cooper returned to the BBC studios for a grilling by Newsnight’s seventh-scariest presenter, Eddie Mair. To a montage of scrolling numbers and bits of miscellaneous computer kit, Mair gave a rather good potted history of the Mersenne primes (so far as is possible in forty-odd seconds), before Curtis Cooper popped up in a square over his left shoulder on the giant video-wall behind him, while what one can only assume were the digits of the quantity under discussion paraded past in the background. Either due to Newsnight’s computer memory being exhausted from storing the prime in its video-wall servers, or owing to some kind of satellite delay, the interview was plagued by a half-second lag, and Mair slightly annoyingly asked Cooper if he was “good with his tax return”, but at least got a good sense out of him of how the search actually got going and continues at his university.

The only BBC outfit to cover the story without wheeling out Cooper for a chat was Radio 4’s The World At One. Martha Kearney pulled the same trick as Today, prefacing their definition of a prime number with the disclaimer “as you will remember” to prevent the listener feeling like an idiot for not doing so. Kearney talked to Marcus du Sautoy, who was of course more interested in explaining a bit of the proper maths behind the patterns (or lack therof) in the primes than the big number itself. Kearney coined perhaps the best explanation of why people embark on this search, “for the sheer game of it”.

In the main, online coverage (which includes New Scientist, Plus Magazine and of course your own Aperiodical) has managed to correctly typeset that tricksy exponential notation and not make any inaccurate statements. Print media, in a post-Leveson world, of course had to tread carefully for fear of upsetting $2^{57,885,161}-1$’s libel lawyers. Of the mainstream newspapers, only the Telegraph and Independent seem to have even covered the story – of course, both gamely illustrated their coverage with a photo of an abacus. Coverage from The Guardian¹, The Times and bastion of science and maths coverage The Sun were all sadly absent.

The Telegraph got off to a bad start with its opening paragraph (“The number, expressed as 2 raised to the 57,885,161 power minus 1, can only be divided by itself and by 1, making it by far the largest prime number ever identified”) but the rest of the article is essentially correct, but slightly annoyingly refers to ‘Mersenne primes’ on more than one occasion without any attempt to define the concept. They also used the word ‘group’ when what they really meant was ‘set’, irritating truly committed mathematical pedants.

While The Telegraph wisely avoided attempting to type the number at all, The Independent fell foul of the gods of superscript and described the number as $257885161-1$ (eagle-eyed readers may be able to spot a divisor). They also used the magnificent “formula of $2p-1$”. Superscript issues aside, their article is very readable and covers all the main points well. It’s also accompanied by a “History of Prime Numbers” section to give some further maths background. Unfortunately it’s marred by a few errors, and while I choose to assume it at least looked pretty as a little fact-box in the dead-tree version, the grim reality of HTML has sadly sucked all the formatting life out of it.

While some of the coverage has been mildly annoying, it’s at least been on the whole largely accurate (Fox News excepted). Discussion continues as to how making a big deal out of this kind of discovery affects the public image of maths – does everyone (and a great deal more than that, if this tweet is to be believed) think it’s boring, and that mathematicians are just stamp collectors? Or is it just good to have maths in any context talked about on the radio and in the papers? The comments thread on this Slate article is full of people having a conversation about mathematics (best quote: “I wonder if a positive density of elliptic curves have Q-rank >0”) which then descends into quantum physics. Of course.

At least, thanks to Euclid, we know the media has another $\aleph_0$ attempts at getting this story right.

Listen Yourself:
BBC World Service – NewsHour (starts around 18:40)
BBC Radio 4 – Today (starts at 1:18:00)
BBC Radio 4 - The World At One (starts at 42:20)
Watch the segment on Newsnight

1. While the main Guardian newspaper had no article that we can find, the story was of course covered by lovely maths expert Alex Bellos (that is to say, an expert on maths who’s lovely, although I’m sure he knows a lot of lovely maths) and in what can only described as psychic journalism, Mersenne numbers were discussed in detail in a prescient and Numberphile-philic blog post by the Guardian’s GrrlScientist fully two days before the discovery was even made, let alone when the news broke.

Katie Steckles: Of course, the most noteworthy thing which happened in 2012 was the creation of an amazing mathematical blogging website, but I don’t mean to go on too much about that. Anyway, we’ve gathered together some candidates for some categories we made up, and will decide on our favourites via the process of arguing.

CP: Allons-y!

Best diagram

The Candidates

Random walks on π
Factor conga
Flat torus
Grow your own food

CP: I’m pretty proud of the graphs I did for Katie’s piece about Herman the German friendship cake, but if we’re honest with ourselves, it’s a two-way race between the walk on π and the factor conga. The walk on π is beautiful, but I’m powerless to resist the jaunty jig of the factor conga.

KS: As a topologist, I feel I should defend the flat torus embedding we reported on in May, along with a lovely video from CP explaining the maths. But secretly I love the factor conga too much.

CP: Let’s all do the conga!

KS: Da-da da da da da, three!

The award goes to…

The factor conga!

Most coincidental numerical coincidence

The Candidates

12:12:12 12/12/12
7:08:09 10/11/12
20:12 20/12/2012
21/12/12 a.k.a 13.0.0.0.0
Leap second

KS: Due to what can only be described as the numerical coincidence that this year’s last two digits are the largest they can be while still being the number of a month, and also of a time on the 12-hour clock, this year has been full of “exciting” and “interesting” dates and times (if you’re a stamp collector). Consecutive number fans will have been excited by 7:08:09 on 10/11/12, whereas fans of repetition had 12:12:12 on 12/12/12, and 20:12 on 20/12/2012 to get themselves het up over. Speaking of getting het up, plenty of people got excited about 13.0.0.0.0 (or, if you’re not familiar with the Mesoamerican Long Count Calendar, 21st December 2012) which was the end of the Mayan calendar and according to some the date of the apocalypse (apocalypse-related satire). However, my personal favourite moment this year was the leap second, which was shoehorned in to everyone’s clock faces at (well, just before) midnight UTC on 30th June 2012, and Peter made a spectacular video of a whole screenful of clocks getting mildly confused by it.

CP: 2012 was the year when I became infected with the Interesting Times Madness. I can’t let a palindromic time pass me by these days without shoving a digital display in the face of someone nearby, with the kind of grin worn by a toddler who’s very proud of his first poo.
Somehow, I managed not to get excited about 12:12:12 12/12/12. Yes, it’s a lot of twelves, but twelve is quite a boring number. Give me a good 17:19 (twin primes) or 12:36 (so many factors!) any day. (You can).
The award’s for the most coincidental coincidence though, so I think I have to go with 7:08:09 10/11/12. The leap second was a real thing! There was nothing coincidental about it!

KS: Well, I agree about it needing to be coincidental; but I don’t think 7:08:09 10/11/12 should take the crown – we’ll get a similar one next year, whereas $12^6$ and its ilk won’t happen again for a huge while. So that.

CP: Righto.

The award goes to…

12:12:12 12/12/12

Most or least plausible press release formula

The Candidates

Christmas trees
Perfect pint
Holiday

KS: We recently featured a guest post by Alistair Bird about a set of formulae for calculating the correct amount of festive tat to throw at your Christmas tree. But it’s not been the only one this year, by a long chalk! The classic format of getting some vaguely qualified mathematical humans to put some symbols in an order, and then claiming it’s a ‘formula’ for something, in order to shift more of your product, has continued to prove popular in 2012 with advertising companies and newspapers looking to fill space. Our favourites included this formula for the perfect pint of beer, and this interesting formula for the perfect holiday, which seems to have used some kind of linear model in several variables to fit a curve to existing data about what makes holidays good – which makes it actually slightly mathsy, and therefore my pick of the bunch.

CP: I’m not sure if we just deleted them all from the news queue and our memories, or if 2012 was a quiet year for nonsense formula stories, but we couldn’t find any real stinkers for this award. I’ll keep an eye out for next year.

The award goes to…

The formula for the perfect holiday!

Most generously worded press release

The Candidates

Has schoolboy genius solved centuries-old problem?
Flea times tables as scientists teach insects how to count
Numerical prodigy sets Guinness record
Brain-damaged college dropout became maths genius after attack

KS: There’s hope for anyone without maths skills, as a man who was attacked by muggers and repeatedly kicked in the head now “sees mathematical formulas and turns them into stunning, intricate diagrams he can draw by hand.” Maths students all over the country are now trying it on with nearby ruffians in the hope of improving their mental powers. Animals having “learned to count” is a classic non-news story, and props to Metro for their excellent pun on the story about fleas evolving number sense. Also: a lovely bit of pseudo-deception from the Vedic ‘numerical prodigy’, who subtracted one insanely long number from another, insisting that in order to make it maximally difficult, the numbers should be chosen so that at every digit a carry was needed. Actually, that made it significantly easier.

CP: However, our winner was clear: The “schoolboy genius” problem took us, and the rest of the Internet, a full week to get to the bottom of. And while it looked like the poor headline was the Daily Mail’s fault, it was the organisers of the competition! Poor show, German competition organisers.

The award goes to…

“Has schoolboy genius solved centuries-old problem?”

Least efficient mathematical instrument

The Candidates

Domputer
QAMA calculator

KS: 2012 has been a great year for inefficient computers – we learned about the QAMA calculator (covered in Episode 93 of the Math/Maths podcast) which won’t tell you the answer to your calculation unless you give it a reasonable estimate first. But our winner must surely be, clocking in at 11.5 microhertz, Matt Parker’s ambitious domino computer at Manchester Science Festival, which took a team of fourteen people over six hours to construct and was able to calculate $4+6=10$ in binary.

CP: …with a failure rate of 50%. Truly magnificent inefficiency.

The award goes to…

The Domputer!

Least or most worthwhile Alan Turing tie-in

The Candidates

The Bletchley Circle
The Turing Enigma
T&F’s free-for-a-very-short-time papers
Turing facebook timeline
Turing machine problems
Turing sunflowers

CP: We saw some pretty shoddy attempts to hitch a ride on the Turing centenary bandwagon. I expunged the worst offenders from our news queue, but of the ones that remain, I think the award has to go to publisher Taylors & Francis for a wonderful bit of rent-seeking. They made a few articles available to read for free, including two by Turing himself (they belong in a museum!),
The Bletchley Circle made a good go of it but was, when we’re honest, a clumsy Turing tie-in.

KS: Speaking of tie-ins, The Turing Enigma was “a dark thriller, commemorating the tragic death of Alan Turing,” and was made available to watch online. Manchester Science Festival did some actual proper science in the name of Turing, getting people from all over to grow sunflowers and send in photos, to be analysed by actual mathematicians and contribute to the body of data on Fibonacci numbers in sunflower spirals. There was also a competition involving Turing machine tape games, in which you had to employ algorithmic problem solving to win prizes. But my favourite Tur-thing of the year was the complete Facebook timeline created by the Science Museum, containing events from Alan Turing’s lifetime – from his birth to the present day. It gives a lovely overview of the events of his lifetime, and then beyond his death it lists all the events held to commemorate his life and work. A bit like this paragraph.

CP: I’m switching sides from Team Misanthropy to Team Positive Mental Attitude – that Facebook timeline was lovely. A deserving winner.

The award goes to…

Alan Turing’s Facebook life timeline

Weirdest story

The Candidates

The strange case of Misha Verbitsky and the trademarked beard
Japanese researchers create a crab-based computer
Advances in pure nonsense

CP: I’m a connoisseur of weird maths. 2012 has been a fruitful year for fruitloop maths stories. The idea of a computer made from crabs gives me a gentle tickle in my brain when I think of it, but Misha Verbitsky’s internment was truly surreal.

KS: However, our most popular weird maths story this year was the story of the ridiculous made-up maths paper which got accepted by a journal, despite having been written by a computer algorithm. The post on our site got picked up by Slashdot, and got nearly two thousand views during October. So, the people have spoken.

CP: Very well. I’d just like to add that Igor Pugach’s beard (KS: also, his website) is truly ridiculous.

The award goes to…

Independent, Negative, Canonically Turing Arrows of Equations and Problems in Applied Formal PDE

Best podcast

The Candidates

Relatively Prime
Math/Maths
The Aperiodcast
Ruled out are Alex Bellos’s Land of the Rising Sums and Matt Parker’s The Turing Solution, for being on lamestream media.

CP: Relatively Prime, no question. If you still haven’t listened to it, make a new year’s resolution to spend a few hours listening through the eight fascinating episodes.

The award goes to…

Relatively Prime!

Highest-ranking mathematician

The Candidates

Sir Tim Gowers
Baronne Daubechies
Mathematical new year’s honours

KS: There were several mathematicians mentioned in the 2013 New Year’s honours, and of course mathematician and Fields Medalist Tim Gowers was made Sir Tim Gowers for the Queen’s birthday honours earlier in the year. Surely that’s the highest honour that’s been bestowed on a mathmo this year?

CP: ‘Tis the season for Lords a-leaping, and Ingrid Daubechies leapfrogged the positively common Sirs and Dames of the mathematical establishment when she was given a barony by the King of Belgium. I think this makes her the highest-ranking mathematician at the moment, unless there’s an emir or a sultan I’m missing.

The award goes to…

Baronne Ingrid Daubechies

Best knitting

The Candidates

Prime number sweater
Conjoined mobius hat
Escher patterns
Spiky icosahedron
Einstein Illusion wall hanging

KS: We’ve featured several mathematical knitting posts this year, including this conjoined Möbius hat, knitted MC Escher designs and my own pattern for a spiky icosahedron. We even went to the MathsJam conference and talked to Pat Ashforth and Steve Plummer about their amazing illusion wall hanging. But my favourite thing this year was Sondra Eklund’s prime number factorisation jumper – nerdy and cosy, it will help you both combat cold weather and factorise numbers.

CP: Glad we agree. Mine’s a size XXL, please.

The award goes to…

The prime number factorisation jumper!

Most or least useful maths website

The Candidates

OEIS
unicodeit.net
detexify
Wolfram|Alpha
sagenb.org
Prime number pooping bear
makebigmaths

KS: We’ve found a variety of useful websites this year – several for LaTeX editing: I find CP’s Make Big Maths really useful for if I need an image of a mathematical symbol I can screen grab; I also like detexify for if I can’t remember the LaTeX command for a symbol, but I can remember what it looks like. As always, we’ve found maths internet staples Wolfram|Alpha and the OEIS incredibly useful (at the MathsJam conference, after being asked a question along the lines ‘how would you calculate this’, one of the shouted replies was indeed ‘Wolfram|Alpha’). We also made use of sagenb.org, to create the graphs for our Grow Your Own Food post (it allows you to use the computer algebra system Sage, in your browser).

CP: Prime number pooping bear prime number pooping bear prime number pooping bear prime number pooping bear prime number pooping bear prime number pooping bear prime number pooping bear prime number pooping bear.
unicodeit.net is a wonderfully simple tool that gives you the corresponding unicode character for any LaTeX command.
But seriously, prime number pooping bear.

KS: Fine. I concede on this one.

The award goes to…

The prime number pooping bear!

Trolliest troll

Uncontested

An upside-down un-award goes to Andrew Hacker, for “Is Algebra Necessary?”, a completely unnecessary article questioning the value of ‘algebra’, recommending instead algebra.

The un-award goes to…

Andrew Hacker!

Award to us for not mentioning the terrible thing all year

CP: Well done us for not sharing our views about the terrible, overpriced, half-baked cash-in that was Everything is Mathematical.

KS: Although it did result in some nice maths puzzles being printed on random pages of The Times, by way of advertising.

CP: And some nice-ish videos done by good people like Kit Yates and Thomas Woolley. But that doesn’t excuse the symbols-floating-in-empty-space intro sequence or the rest of the entirely derivative, lacklustre tie-in site. Which hasn’t seen an update since September, by the way.

Most ubiquitous popular mathematician (The du Sautoy Prize for Omnipresence)

The Candidates

Keith Devlin
Matt Parker
Dara O Briain
James Grime

CP: Marcus du Sautoy is of course barred from this award due to the fact he would have won it if we’d been doing these awards in any of the past 10 years. He remains the media’s go-to maths talking-head, but to give him this award would only serve to encourage him.

KS: Various people have been doing their best to shove maths in the general public’s faces this year, not least of which is stand-up and physics graduate Dara O Briain, who somehow managed to get a maths programme commissioned on TV’s creative wasteland Dave. Dara O Briain’s School of Hard Sums was aired this year, and featured Marcus Du Sautoy along with a selection of comedy guests and a sprinkling of maths undergrads sitting in the corner, and managed to present a selection of puzzles ranging from mildly trivial to good fun, while simultaneously both reinforcing and challenging the stereotypes about mathematicians.

CP: You’d think Keith Devlin has been around doing popular maths stuff long enough that he’d sort of fade into the background, but that thankfully isn’t the case. As well as his long-running MAA column Devlin’s Angle, Prof Devlin has been popping up all over the place to talk about his Introduction to mathematical thinking MOOC and new book. Keith has become NPR’s resident Math Guy, called on to talk about whatever maths is fit for the radio. Unfortunately for the good professor, he’s based in the US so we only get a smattering of his output, which means he was pipped to the post by…

KS: Friend of the Aperiodical Matt Parker, whose main job is to talk about maths and entertain people, has been doing both with aplomb this year – from a comedy engineering show at the Edinburgh festival, to appearances on both TV and radio in documentaries about Alan Turing. He’s also toured with Festival of the Spoken Nerd, created a computer out of dominoes for Manchester Science Festival, and broken the world record for most simultaneous Rubik’s cubes being solved while raising money for charity. He also appears in this promo video for James May’s new set of YouTube channels, representing the maths strand.
But my vote for most ubiquitous popular mathematician goes to internet pixie James Grime, who’s done hundreds of YouTube videos (both for his own channel and for Numberphile) while keeping up his day job of talking about the Enigma machine all over the country. He’s also appeared on the radio talking about maths, including the Today programme and stats show More or Less, and been hilariously entertaining on Twitter.

CP: “Internet pixie”? Yes, I suppose James does keep popping up everywhere. And here he is again, winning an award.

The award goes to…

James Grime!

Well, that’s it for this year! If you’d like to see yourself in this list in a year’s time, make sure you do something crazy/impressive/fantastic enough to get on our radar. We look forward to it.

Trophy photo by Wikipedia user FFahm, CC-BY-SA licensed.

Exactly six months ago, we launched The Aperiodical. Since then, we’ve published 523 posts to 115,000 visitors; been slashdotted, Hacker Newsed, and reddited; mentioned on Radio 4; got to the bottom of a mystery; been inordinately proud of a new set of fonts; published pieces by 11 guest authors; and laughed all the way to the bank. Except the last one.

We thought we’d take this opportunity to gather together some of the best bits of the first six months of this venture, and reflect on what’s gone wrong and what’s gone right.

Some of the best bits

Electoral reforms and non-transitive dice by Andrew Taylor

Published before the site officially launched, but it’s great, and was mentioned on Radio 4’s Material World, where we were referred to as “the journal Aperiodical”!

Klein: outside the bottle by Peter, Christian and Katie

Blood, sweat and tears were shed writing this piece, so we could launch the site with a bang. Amazingly, only one pedant on Reddit picked us up for saying that the Klein bottle is 4-dimensional. Nothing we’ve written since has been proofread as closely as this.

We also posted a couple of other Klein-related items for the launch, including a video made by Matt Parker specially for the Aperiodical, outlining the Top N Facts about the Klein Bottle.

Classic maths books reset with LaTeX on Project Gutenberg by Christian

This was just a little news story posted the day after we launched. It was picked up by the Hacker News hordes and the traffic caused the server to fall over pretty much instantly. Lesson learned: turn caching on!

Grow your own Food by Christian and Katie

Katie and CP collaborated on this thought experiment taking Herman, the German Friendship Cake to his logical conclusion.

In what flipping dimension is a square peg in a round hole just as good as a round peg in a square hole? by Colm Mulcahy

Colm Mulcahy emailed offering to write some posts for us very soon after we launched, so we gave him a column straight away. He’s written some great stuff, including this great piece about a very unintuitive geometrical fact.

Has schoolboy genius solved problem that baffled mathematicians for centuries? by Peter and Christian

A few newspapers reported that a schoolboy had solved a problem “posed by Isaac Newton” which had stood unsolved “for over 300 years”. We were pretty skeptical, and over the course of a month we managed to get to the bottom of it.

Historical anniversaries: are they worth celebrating? and Mathematics: a culture of historical inaccuracy? by Peter

A pair of posts Peter wrote before the summer about how we view history. The first, on the eve of the Turing centenary, wondered whether we should be as excited as we seem to be about anniversaries. The second considers whether those anecdotes we know aren’t really true are a valuable part of mathematical folklore and should be encouraged.

WLTM real number. Must be normal and enjoy long walks on the plane by Christian

CP enjoyed writing this one, and got some nice comments from the authors of the paper!

Turing Round Up by Christian

Turing-mania reached fever pitch this year as we approached his centenary. We decided to collect all our Turing news into one easily-manageable post, both to save our sanity and your RSS reader’s. People made some great things to celebrate Alan Turing Year, but separating the wheat from the chaff was quite hard work.

To teach, must I principally research? and “I’m not a mathematician, the maths I’m doing is really just basic modelling” by Peter

A pair of posts Peter wrote in the run up to the start of university term, the first about the place of non-researchers in providing university education and the second about the mathematical level of work undertaken by graduates in employment.

Regular Features

Part of the reasoning behind the naming of the site was that we weren’t planning on having too many regular features, but we seem to have accumulated quite a few over the six months we’ve been going.

Puzzlebomb

Katie’s regular monthly puzzle sheet, which already existed on Twitter, was adopted by the Aperiodical as a regular feature. While the puzzles don’t always involve maths directly, they enjoy having a home on the site. In addition, a couple of the puzzles which required some maths to construct have been given nice write-ups on the site by puzzle fiend Paul.

Carnival of Mathematics

This blog carnival has been running for several years, and Peter became aware it was in need of a new caretaker – previous Carnival boss Mike Croucher was looking for his spare time back – so the Aperiodical adopted it and now coordinates the monthly round-up of maths blogging, appointing hosts and publicising each month’s edition. (Future volunteers to host are always welcome!)

Aperiodcast

After the site had been running for a while, we realised we weren’t using enough puns on the word ‘aperiodical’, so we started the Aperiodcast – a completely non-regular audio podcast of recent posts we’d enjoyed and that had been popular, in amongst our random squabblings and geeking out about fonts.

MathsJam writeups

Maths Jam is a monthly meeting of like-minded self-confessed maths enthusiasts who get together in a pub to share stuff they like (strictly limited to “Puzzles, games, problems, or just anything they think is cool or interesting”). Sometimes, still buzzing from the excitement of the previous night’s Jamming, we’ll feature a writeup of a Maths Jam night – often in one of Katie, CP or Peter’s homes of Manchester, Newcastle or Nottingham.

Our thoughts

It hardly seems five minutes since Peter Krautzberger of mathblogging.org called The Aperiodical the “clear contender for single best thing that happened to math — sorry, maths blogging this year“, but actually that was only two weeks after we officially launched and that was six months ago. What do Katie, Christian and Peter think about the experience?

Katie

It doesn’t feel like six months has passed already! The site has become (I reckon) a really nice collection of interesting things, and if I weren’t already involved I’d certainly have it as one of my RSS feeds. I’m also pleased with how we’ve integrated it with Facebook and Twitter, as it means people can share things really easily and we get a conversation going. It’d be nice to get some more good feature posts, and I hope to convince a couple more (exciting) authors to start guest columns. I’m looking forward to see how the site grows and what happens next!

Christian

First of all, I’m really pleased with how the site has developed. It’s been quite successful in meeting the aims we set out for it, and I’ve thoroughly enjoyed running it with Peter and Katie.

A couple of things haven’t worked out: we’ve slowed down on the long, well-researched pieces like the Klein one, and we haven’t had as many guest authors as we wanted. I think we all realised that we didn’t have enough time to keep up with posting news stories, run the various regular features, and write in-depth pieces while holding down our day jobs.

While the pieces we have published from other authors have all been great, there haven’t been many of them. There are loads of people writing great stuff on their own blogs, so maybe it was arrogant to think we could just ask them to post on ours instead, or maybe we could do some more to drum up pieces for the site. For a while we considered applying for grants to pay for pieces, but we decided that would lead us down the road to icky commercialism.

The things that have worked outweigh those two failings by far. I’ve really enjoyed writing not just my “regular” columns (and I really do enjoy writing the Aperiodical Round Up), but keeping abreast of news and doing research into stories or new papers has been much more interesting than I thought it would be.
I want to do more with the look of the site in future. At the moment we’re using the bloated-yet-competent Pagelines theme for WordPress, with a few customisations and the beeeeautiful Computer Modern fonts. I think we’ve mentioned how pleased we are with the fonts. Producing graphics for stories is very time-consuming but I’ve come to realise how much better they make everything look.

Peter

What can I say? The Aperiodical is an unpaid commitment bringing hassle and obligation into my already busy life and distracting me from the far more important things I should be doing.
Can we stop now?

Finally, thanks to Andrew and Paul Taylor, Alistair Bird, Samuel Hansen, Matt Parker, Steve Mould, Johnathan Gregg, Julia Collins, Madeleine Shepherd, Colin Beveridge and Colm Mulcahy for writing some really excellent stuff and giving it to us for free. Much appreciated!

While Emily had no formal composition training until postgraduate level, she did have formal training in mathematics, and she feels that this has informed her composition style. “It was natural to use techniques from maths in my composition.” Her work incorporates mathematical functions such as exponentials, random numbers and layered combinations of these. As well as helping her to decide the overall structure of the piece, such as where the dramatic moments will be, it can also specify aspects of the music on a micro level, such as rhythm, pitch and which section of the orchestra is playing. “Composition is all about making decisions.” Emily’s choice to use mathematical techniques often helps to make those decisions for her – to the point that she sometimes uses computer programs which speed up areas of the compositional process.

“Calculus of the Nervous System” was a phrase used by Ada Lovelace to describe her attempt to create a mathematical model for how nerve impulses give rise to thoughts and feelings in the brain. Ada, the daughter of Lord Byron, worked with Charles Babbage on his difference engine and was instrumental in documenting the construction of the engine. Howard’s composition, inspired by Lovelace, brings together many of her mathematical function composition techniques, as well as incorporating aspects of neural networks, and is constructed as a sequence of memories, some more clear than others, which overlap and interact: “like a muddled brain”.

This isn’t the first piece inspired by Lovelace that Emily has written; her ‘Ada Sketches’, written for mezzo-soprano, flute, clarinet and percussion, is a short dramatic vocal work in which Ada ‐ immersed in calculations – imagines that the analytical engine might compose music. ”Ada said, ‘Suppose these numbers represented musical notes; we could feed them into the engine and see what comes out’. This was very prescient of her – she was years ahead of electronic music and computer generated music.” Emily hopes that ‘Ada Sketches’ , which incorporates Ada’s voice as she works with the engine, and aspects of her character, as well as the engine itself being present and ‘speaking’ in the music, might form the basis of a future opera.

Emily is also mentoring a pilot project with students from the Bridge Academy, a school in Hackney, London which has specialisms in both mathematics and music. “The year 9s have been working with members of the London Symphony Orchestra to develop a composition based on mathematical principles facilitated by myself and Michael Betteridge, a postgraduate composer studying at the Royal Northern College of Music. The mathematics involved in their composition is the mathematics they’re learning at school.” The composition resulting from this week-long creative project was performed at a school concert by members of the London Symphony Orchestra alongside the students who composed the work. This sounds like a fantastic way to get the kids interested in both maths and music, and I hope that as many are inspired by it to go on and become mathematicians as composers.

Emily’s website lists all of her compositions, many of which are inspired by mathematical and scientific concepts, as well as information about future performances of her works. Do you have both a musical and a mathematical side? What’s your favourite mathematically-inspired piece of music? Comments below.

It’s an unpresupposing little letter, $x$. In fact, that’s the reason we use it to represent something we don’t know. But how do you write it down? When Vijay Krishnan tweeted a link to an American college professor’s page on mathematical handwriting, I was shocked to learn that he thought adding a hook to a simple cross was sufficient to differentiate letter-$x$ from times-$\times$.

So I asked our Twitter followers how they write $x$. The Cambrian explosion of diversity in answers I received was eye-opening - I’m glad I asked!

I’ve collected the tweets together in a story on Storify.

Click here to read the story.

In summary, it seems that no. 3 is prevalent only in the British Commonwealth and Russia, while no. 2 only appears in the US. Continental Europeans seem to use no. 1, though the calligraphic capital $\mathcal{X}$ caused some confusion. Commentators on either side of the Atlantic were aghast at each other’s orthography, and thought the reasons for their own methods were self-evident. I think we spread a little understanding in the end though.

As a postscript, I have to mention the reference for all things to do with mathematical notation, Cajori’s A History of Mathematical Notations¹. In particular, Cajori tells us that it was probably Descartes who introduced the use of $x$ (and $y$ and $z$) to represent unknowns.

Sadly, it doesn’t look like Cajori concerns himself with handwritten maths after the adoption of printing in Europe, so I don’t know where or when the British/Russian $x$ first appeared. Florian Cajori: lightweight scholar!

And that’s my final word on the letter $x$.

1. Warning: if you start reading Cajori, the Sun might go up and down a few times before you stop reading Cajori.

Steve Mould

There are $2\pi$ radians in $1$ whole turn. With $\tau$ you get $1 \tau$ radians in $1$ whole turn. Much easier.

But what about Euler’s identity? Well that becomes:

\[e^{i\tau} = 1\]

Beautiful. And look at these:

\[\begin{align} e^{i \frac{\tau}{4}} &= i \\ e^{i \frac{\tau}{2}} &= -1 \\ e^{i \frac{3 \tau}{4}} &= -i \\e^{i\tau} &= 1 \end{align} \]

See how the factor of $\frac{1}{4}$ represents $\frac{1}{4}$ of a turn through the complex plane and so on. With $\pi$ that’s hidden.

The area of a circle becomes $\frac{\tau}{2} r^2$. Surely $\pi$ would be better here? Not at all. The half tells us that we’ve performed an integration. With $\tau$, unexpected terms tend to point you in interesting directions. With $\pi$ they’re obscured.

Sticking with $\pi$ is just irrational.

Matt Parker

If $\tau$ is so great why did we even have $\pi$ in the first place? Why have we used $\pi$ for millennia instead of switching to $\tau$ if it’s so great? Because from the very first circle-measurements of prehistory to modern precision engineering: the direct measurement you can make of a circle is its diameter, not its radius. Comparing the ratio of diameter to circumference is more fundamental.

$\tau$ evangelists can wheel out as many formulae and equations as they want containing $2\pi$, they are just cherry-picking. There are just as many equations using $\pi$, including the immortal $e^{i\pi} + 1 = 0$. If you convert Euler’s Identity to use $\tau$, you no longer have the irrational exponential constant, the transcendental circle constant, imaginary numbers, unity, negative numbers and zero all linked together in one relationship.

Of course the final word in this argument will come down to radians. $\pi$ radians gives you half a circle, $\tau$ radians gives you a whole circle. $\pi$ is the more nuanced base-unit; it actually divides up what it is measuring. $\tau$ radians will bring you right back to where you started; with $\pi$ radians you actually get somewhere.

Festival of the Spoken Nerd will be appearing at this year’s Edinburgh Festival from 1-7 August, and touring the UK in October and November. For more details, visit FOTSN.com.