The next issue of the Carnival of Mathematics, rounding up blog posts from the month of April, and compiled by Becky, is now online at Lines, Curves, Spirals.

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>On the day, we’ll be editing from **10am-3pm BST** (although we don’t expect everyone to edit for the whole day, or even to stay within these times – we just want to give people a nudge/opportunity/excuse to get involved). You’ll need to be somewhere with a computer and internet connection.

If you’re near to one of our real-world meets, you could join in there – Katie will be editing in Manchester at Google Digital Garage, where she’s reserved a room, and Nicholas Jackson is happy to join people in Coventry. Check the shared Google Doc for details of how to contact them, and to add your own real-world locations if you like.

To edit pages on Wikimedia, you don’t technically need an account – edits can be made using the IP of your computer as an identifier – but it’s better to have an account, so you can keep track of your edits, respond to questions, and not have your edits confused with those of others who happen to be on the same wifi network (and like to put rude words into things and get your IP blocked/banned). You can sign up for an account here.

It may also be worth doing some research to find suitable quotes, or lists of people that might be a good source of quotes – although it’s probably fine to leave this until the day, as we’ll hopefully have time to do this then. We’ve made a start in the shared Google Doc.

Our main plan is to fix up the Wikiquote Mathematics page – to that end, we’ve started collecting quotes in the shared Google Doc. Each will need to be separately verified from a second source, by someone other than the person who suggested them, and then can be added to the Wikiquote page. You can also find more examples of quotes to add to the doc, and make sure you’re clear about who’s going to do which bit by putting your name next to the task in the shared doc (and then making sure you do it!)

Editing Wiki pages involves knowing a little bit of markup – most of the knowledge you’ll need can be gleaned by looking at the existing entries on the page, and if you make any mistakes someone can always fix them (it is a Wiki, after all, so someone probably will anyway before too long). If you’d like more in-depth info, there’s a great how-to guide on Wikiquote, with a list of markup.

Once we’ve pimped out the Wikiquote Mathematics page, there are other pages within Wikiquote, and elsewhere on Wikipedia, that can also be worked on while we’re at it. We’ll use the shared Google Doc to suggest things for people to work on, and it’s totally up to you what you get involved with. We’ll also aim to run a video Hangout, for which we’ll post the link in the shared Google Doc, so that people can chat to each other (if they’re not somewhere that would inconvenience anyone).

We hope you can join us to help improve Wikipedia for female mathematicians, and that you can learn a little about editing Wikipedia while you’re at it – it would be great if you can carry on doing this work in your own time afterwards too.

If you have any questions about the event, or what to do, check the shared Google Doc, or email katie@aperiodical.com and we can try to help!

]]>My first two posts are:

- a writeup of Robert Langlands’ work that won him this year’s Abel prize
- a post expanding on the recent chromatic number result by Aubrey de Grey

Keep an eye on the Spektrum blog, and the Aperiodical Twitter feed for news of further posts!

]]>I’ve made another one of my interactive online maths doodads. You should have a go at it right now. It doesn’t require any effort on your part, other than coming up with a positive integer.

Isn’t that incredible?

The trick is based on the paper “Every positive integer is a sum of three palindromes” by Javier Cilleruelo, Florian Luca and Lewis Baxter. That’s a superb fact, and one that was very hard to prove. And the authors didn’t just do it in base 10: they show that it’s true for any base from 5 upwards.

Cor!

So obviously, when I discovered this theorem, I had to try it out for myself, and I immediately thought it would make a great interactive toy. The problem is that the paper is forbiddingly complicated. As easy as the theorem statement is to understand, the proof is fiddly in equal measure.

The proof consists of an algorithm which takes any positive integer, and is shown to produce three palindromes which sum to that integer. For numbers 7 digits or larger, the algorithm’s moderately straightforward. However, for smaller numbers, it’s a horrendous proof by cases that really tested my motivation to see the whole thing through.

It took me a couple of days of snatched time between work and parenting duties to implement the whole algorithm in JavaScript, and then another night and a shower before I tracked down the missing case in Algorithm I.3 ($\delta_{m-1}=0$ *is* allowed, and I’m going to stick by that!).

While in the depths of will-it-ever-work despair, I found out that one of the co-authors, Lewis Baxter, has put online a page that performs the trick for you, just like I was intending to do. That was very helpful when I was trying to work out whether I or the paper had gone wrong, but three days in I was in no mood to quit. So I didn’t quit, and finally it looked like I’d implemented the algorithm correctly.

Now, proofs that claim to work for all $n$ are well and good, but I wanted to check at least the “small numbers” that had such fiddly solutions. So, I set my computer off running the algorithm on every number up to 1000000, and on a few hundred thousand randomly-chosen larger ones.

And it worked!

I decided this incredible theorem deserves a hype man, so everyone knows just how mind-blowingly cool it is. I spiffed up my web interface with bright colours, a lot of patter, and an ostentatious amount of whizzy text. A moron in a hurry will be in no doubt that this is a very cool result.

Try the incredible palindromic hat-trick now.

*I hope I didn’t overdo it with the whizzy text.*

In geometric graph theory, the Hadwiger–Nelson problem poses the question: what is the minimum number of colours required to colour the 2-dimensional plane, so that no two points that are exactly one unit of distance apart are coloured the same colour? The answer to this question is called the **chromatic number** of the plane.

Up until now, we knew the answer was one of 4, 5, 6 or 7. The diagram shown here contains proofs of the upper and lower bounds: for the upper bound, you can assume the hexagons are each less than one unit in diameter, and so each point inside a hexagon is one unit away from a circle of points which use only the other six colours, as the colouring has no adjacent or adjacent-but-one hexagons the same colour.

The graph in the diagram also proves the minimum is four: known as the **Moser Spindle**, the graph consists of two pairs of equilateral triangles, and all the arcs are one unit long. It can be used to prove a three-colouring is not possible, as follows. The colour of the top left vertex forces the other two vertices of each of the two attached triangles to use the other two colours, but this in turn forces the two lower right vertices to be the same colour, and they are a unit apart. Contradiction!

Aubrey de Grey’s proof, uploaded to the arXiv earlier this month, uses a similar line of reasoning to prove it’s not possible to colour the plane with four colours in this way – only this graph has around 1600 vertices (see diagram, if you can), and the construction and checking of the graph is computer-assisted. The paper itself is still just on the arXiv for now as far as I can tell, but if we spot it being peer-reviewed and published anywhere we’ll make an update here.

In the wake of this announcement, a 16th Polymath project has been proposed to try to find a simpler graph, and improve on this result – some have conjectured it might not be possible to reduce the size of this graph further without redesigning it completely, as de Grey’s method already reduces it to as minimal as possible from the initial approach used (it involves taking smaller graphs and rotating them to align on top, much like the Moser Spindle).

Some blog posts on the topic, including one which makes attempts to verify the proof, and the paper itself, are listed below.

The chromatic number of the plane is at least 5, de Grey’s paper on the ArXiv

The chromatic number of the plane is at least 5, on Jordan Ellenberg’s blog

Amazing progress on longstanding open problems, on Scott Aaronson’s blog

The chromatic number of the plane is at least 5, on Dustin Mixon’s blog

Aubrey de Grey: The chromatic number of the plane is at least 5, on Gil Kalai’s blog

For a while now I’ve been fascinated by the story of Claude Shannon, the pioneer of information theory and the originator of many fundamental concepts now used in all modern manipulation and transmission of data. Being sent a copy of this biography to review was a great chance to find out more about his work and life.

**A Mind At Play: How Claude Shannon Invented the Information Age
**

The authors, who describe themselves as biographers and writers foremost, have taught themselves the mathematics they need to explain Shannon’s work, and weave in some excellent and succinct explanations of the concepts amongst a fascinating human story. From his early years as an enthusiastic maker and tinkerer, through his various university courses and his placement at Bell Labs, to his later years at MIT and retirement, Shannon’s life is chronicled in detail, with a spread of well-chosen photographs to accompany the story.

Claude Shannon is described as the father of information theory – his seminal 1948 paper outlined concepts including the fundamental nature of binary numbers (coining the word ‘bit’, a binary digit), information density, communication channels, and the theoretical Shannon Limit of how quickly digital information can theoretically be transmitted in a noisy channel. These ideas predated even simple computing machines, and Shannon’s work was perfectly timed to provide a foundation for those creating early computers.

The story gives a real sense of how Shannon was well placed to create the mathematics he did – with a sharp intellect that was torn between his love of abstract mathematical theory and his fondness for hands-on inventing and engineering, he had just the right mindset to see what communication theory would become and how it could be made rigorous in a mathematical framework.

It’s also fascinating to learn about Shannon’s other passions in life – nothing he did before or since comes close to the major impact his work on information theory had, but it was far from his only passion. Other areas of mathematics and engineering, as well as pastimes including juggling, stock market predictions, and building robots all fell to his mighty intellect and he brought huge joy to the people around him with his stories and ideas.

The book is well written and lovingly put together (and has a frankly beautiful cover in the hardback edition). It was enjoyable to read, and full of interesting facts and stories. I didn’t realise until reading this book that a wooden box I have at home, which has a switch on top that when flipped, engages a robotic arm that pops out and flips the switch back again, is a modern incarnation of an invention of Shannon’s – he called it ‘the ultimate machine’, one which switches itself off. Knowing this was his original creation, and the joy I find in it, gives me a real sense of connection to this brilliant mathematician whose work changed the world for all of us.

A Mind At Play: How Claude Shannon Invented the Information Age by Jimmy Soni and Rob Goodman is published by Simon and Schuster.

]]>The next issue of the Carnival of Mathematics, rounding up blog posts from the month of March, and compiled by Robin, is now online at Theorem of the Day.

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>Here’s the final set of photos and video clips from the last week, and for the data fiends among you, a sneaky look at my spreadsheet of runs. With a graph, as requested by Hannah Fry.

Day 23 (aka @sportrelief day): stupid bloody GPS watch failed to log my distance, but I fixed it using the screen grab from my route planning map (which I won't share because it gives away where I live). A good pace! #pikmdotrun pic.twitter.com/6YWz03EeEl

— Katie Steckles (@stecks) March 23, 2018

Next up was this stroke of genius:

Day 25: had the genius idea of getting dropped of 3.14km away from the pub, where we're having a Sunday roast #winning #pikmdotrun pic.twitter.com/kndBgPY2gB

— Katie Steckles (@stecks) March 25, 2018

Augh! Just realised my tweet from yesterday didn't send! Day 27: ran out in the countryside near my nice hotel, missed the turning, had to just run 1.57km away and turn round. #pikmdotrun pic.twitter.com/62pzulsbUZ

— Katie Steckles (@stecks) March 28, 2018

Day 30: Final gym run (will be outdoors tomorrow). Accompanied by @aPaulTaylor & Waterbot. https://t.co/dtKU5Rj0qH pic.twitter.com/rm8AGaJqDQ

— Katie Steckles (@stecks) March 30, 2018

This final run video shows clips and photos from a bunch of the days, plus my triumphant final approach on Day 31:

There’s still time to chuck a final £3.14 on the pile at pikm.run, if you haven’t already. Thanks again to everyone! Now I’m going for a sit down.

]]>Extension and abstraction without apparent direction or purpose is fundamental to the discipline. Applicability is not the reason we work, and plenty that is not applicable contributes to the beauty and magnificence of our subject.

– Peter Rowlett, “The unplanned impact of mathematics”, Nature 475, 2011, pp. 166-169.Trying to solve real-world problems, researchers often discover that the tools they need were developed years, decades or even centuries earlier by mathematicians with no prospect of, or care for, applicability.

– Peter Rowlett, “The unplanned impact of mathematics”, Nature 475, 2011, pp. 166-169.There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.

Peter Rowlett, “The unplanned impact of mathematics”, Nature 475, 2011, pp. 166-169.

Now, don’t get me wrong. I have every admiration for Peter and his work; his is a thoughtful voice of reason, and it’s not at all unreasonable for the Wikiquote page on mathematics to cite his writing.

It does seem unreasonable, though, that if a single paper by Peter merits three entries in the list, that the whole of womankind, from the whole of written history, only (currently) merits four (one from Hannah Arendt, one from Iris Murdoch, and two from Simone Weil under ‘quotes that mention mathematics’.)

Incidentally, I’m aware of the gender imbalance in Quotable Maths, which I curate at my blog. Of the last 110 quotes, only 16 are from women. While that’s not as bad as the Wikiquote ratio, it’s still not good enough; I’m working to improve that.

I would imagine, but haven’t checked, that there is a corresponding disparity in the number of quotes by mathematicians of colour, disabled mathematicians and LGBTQ+ mathematicians.

This isn’t ‘Nam, of course, there are rules – sadly, we can’t just jot down a selection of Katie’s carefully-crafted quips and spam the quotes page with them. According to the discussion page:

The quote ought to be understandable outside of its original context (or the context ought to be well enough known that most people (aquainted with the subject matter) will still understand it.

The quote ought to be one of the following:

- Known by many people.
- Uttered by a famous person.

NB: Both of the above need to be tempered by the context of the quote. I.e a quote need only be known by many people familiar with its subject matter.

The quote ought to be one of the following:- Interesting
- Funny
- Rude
- Of significance on its own
- Of significance given its source or some additional context

There must be dozens of those, right? What do you think about finding them and coordinating an effort to redress the balance?

I am not a Wikipedian, so you are at least as much of an expert in this as I am. Luckily, Katie and Peter have put together a Google Doc with the necessary details. We’ll be using this doc to coordinate real-life meet-ups and to document and discuss quotes that merit inclusion. We’ll also set up a Google Hangout for real-time coordination – please email Katie if you’d like to have the link sent to you; it will also be listed in the doc.

Editing wikis is straightforward. You don’t need to be a historian or a mathematician (although everyone is) to take part, you just have to be willing to roll up your sleeves and put some quotes in the right place. If you want to get started on the project ahead of time, it’d be helpful if you could find and add quotes to the Google doc ahead of time, so we have material to work with out of the gate.

So, I look forward to seeing you on **Saturday, May 12th from 10am** to help improve the Wikirepresentation of women in maths. No excuses! As Florence Nightingale herself said:

I attribute my success to this – I never gave or took any excuse.

WikiQuote: Mathematics

How to run an editathon

The Google Doc

Quotable maths at Flying Colours Maths

If I can make it to £1000 before the end of the month, I’ll be pretty pleased! Donate at pikm.run, or see below for my daily sweaty photos/videos/instagram posts.

Day 9: yep, I'm still doing this #pikmdotrun pic.twitter.com/MpFQvEImgr

— Katie Steckles (@stecks) March 9, 2018

On the 9th, I thought I’d make use of the mathematical properties of π to do a slightly silly one, and made a video:

Day 11: gym again. If you missed it yesterday, here's my video from Day 10: https://t.co/rdiPCJ534N pic.twitter.com/knyzH4L6f9

— Katie Steckles (@stecks) March 11, 2018

Day 13: still going. Thanks to everyone who's supported so far! https://t.co/dtKU5Rj0qH pic.twitter.com/p5qUUNncSD

— Katie Steckles (@stecks) March 13, 2018

I also managed to get in a few more joint runs with running companions:

Day 15: now officially kinda halfway! Buddy gym run again with @elsie_m_ #pikmdotrun pic.twitter.com/VhcVIsghll

— Katie Steckles (@stecks) March 15, 2018

Day 17: logistically complex. Managed to run πkm fully inside the B'ham NEC, as I've been working at @BigBangFair today. GPS watch gave up after 2.04km (no signal). #pikmdotrun pic.twitter.com/eebrhKjUgh

— Katie Steckles (@stecks) March 17, 2018

Day 19: staying in the gym due to the cold weather, on the world's shiniest treadmill #pikmdotrun pic.twitter.com/WO9xaPKaSV

— Katie Steckles (@stecks) March 19, 2018

I was also given an amazing gift by maths/knitting fan Linda Pollard, who came to see me perform at a show. She’s written up the knit of these magnificent π/sum gloves on a Ravelry page. I took the opportunity to test out their warmness on my next outdoor run:

This mild cry for help resulted in plenty of nice replies on Twitter, which has been a real boost – including Aperiodichum Colin Beveridge, who pointed out that my total is around $\pi^4$, a pleasing coincidence.

Day 21: Gym again (but went to a different gym for variety). Found it hard today. Encouragement please. #pikmdotrun pic.twitter.com/lIvuC7yBUZ

— Katie Steckles (@stecks) March 21, 2018

And of course, today’s effort:

Running continues. Watch this space for a final wrap-up and fundraising total at the end of the month.

Katie’s fundraising page at Sport Relief

More information about Sport Relief