Chris Watson has written in to tell us about his site, *Tessellation Art*, where he sells his heavily Escher-inspired prints. They’re available in a range of sizes and media, and quite affordably priced. I particularly like the print above, titled *Vortex*.

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

– W. S. Anglin

A few years ago I saw a post on a website that showed that the inverse of 998,001 produces a decimal expansion that counts, using three digit strings, from 000 to 997 without error.

\[ \frac{1}{998,\!001} = 0.0000010020030040050060070080090100110120130\ldots \]

I immediately thought that this had to be a hoax. I decided to work it out to prove it was a hoax – after all some people put anything they want on the web whether it is true or not.

My calculator cannot handle more than eight digits. And I really did not want to work it out by hand. The arithmetic is simple enough, I just did not want to spend that much time on it (we all like instant everything – personally I get impatient waiting on the microwave). So I took this problem to a wonderful website, Wolfram|Alpha. It did the calculation for me, and I checked it out – digit by digit. I did not like what I found out – the equation was right, and I was wrong.

But I did start wondering – I wondered if there were other numbers that had similar properties as this one. Are there other integers that, when you turn the number into a unit fraction and then convert it to a decimal, produce an interesting sequence of numbers (like counting, or multiplying, or whatever).

So I looked on the web for answers – but the internet did not have any answers to this question (stupid internet!). Could it be true that the example I found in a blog post was the only example of this special property?

I contacted a few other mathematicians by e-mail to see if they knew anything about this kind of a mathematics – most did not – but a few gave me so hints on where I might find out more. But those websites only gave me more hints.

By this time I realized two things – I was not going to find an answer out there – and if I wanted to know the answer I was going to have to find it myself. And I really wanted to know the answer!

Things were slow at first, and at second, and at third… but I found a few more hints, and eventually found another number with a similar properties.

Then I found a few more.

But I did not understand why some numbers worked and others did not. So I kept working on the problem, and as I found more of these special numbers I started to see patterns, and I learned how to fill in some blank spots in these patterns.

I decided this group of numbers needed a special name – after all I could not keep calling them just “special numbers. “Mr. B’s Super-duper-fantastic Arithmetically Accurate and Mathematically Mystical Sequence Numbers”. This name is really kind of long and it did not catch on. Now I just call them “Sequence Numbers” because they produce interesting sequences.

I define sequence numbers as integers that produce interesting sequences when you take their inverses and convert them into decimal form. I prefer that the “interesting sequence” is listed in the Online Encyclopedia of Integer Sequences, but if the sequence is an obvious or well known sequence I can accept that too.

So what are some of these other numbers? Let me tell you about some of my new buddies:

- 9,899 produces the first few terms of the well known Fibonacci sequence.

- 196,020 produces a list of the multiples of five.

- 998,999 produces the terms of the Fibonacci sequence, using three digit strings.

\[ \frac{1}{998999} = 0.000 \; 001 \; 002 \; 003 \; 005 \; 008 \; 013 \; 021\ldots \]

- 998,998 produces the terms of the Jacobsthal sequence, using three digit strings.

\[ \frac{1}{998998} = 0.000 \; 001 \; 001 \; 003 \; 005 \; 011 \; 021 \; 043 \; 085 \; 171\ldots \]

- 76,922,923,077 counts by multiples of 13. (Imagine that! How far can you count by multiples of 13?)

- 999,999,700,000,029,999,999 shows a list of all of the triangular numbers with 6 digits or less, and even most of the triangular numbers with seven

- 999,999,999,997 lists all of the powers of two ($2^0$, $2^1$, $2^2$, $2^3$, etc.) up to 12 digits long, except for the largest 12-digit power of 2. And it writes them all in 12-digit strings.

- 999,999,998,999,999,998,999,999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999,999 shows the terms of the tridecanacci sequence, defined as $a(n) = \begin{cases}

0 & 0 \leq n \lt 12, \\

1 & n=12, \\

\sum_{i=1}^{13}a(n-i) & n \gt 12

\end{cases}$

- 999,999,999,876,543,210 – this one shows the 123,456,789 times table (all of the multiples of 123,456,789) up to, but not including the last 24 digit multiple of 123,456,789.

There are a lot of these special numbers out there, and I know I have not found them all.

I hope some of you don’t believe me. Well let me rephrase that – I hope that some of you don’t believe me AND have the gumption to try to prove me wrong. If you do you will learn how to find these numbers, and even how to design some to do specific things.

If you find something you want to show me, or you want me to post for others, please email me at: mbiom.edu@gmail.com. Check out my website at: SequenceNumbers.blogspot.com.

Don’t be afraid of the big numbers. That may be where all the really interesting and fun stuff starts.

]]>But what I really want to tell you about is the National Museum of Mathematics in New York. We couldn’t fly all the way to the East coast of America and not pay a visit. So we did!

MoMath, as it’s known, is more of an exploratorium than a museum – a fairly small space filled with toys and gizmos and whatevers to show rather than tell visitors about interesting mathematical concepts. It’s definitely aimed at kids, of the smallish sort.

Most of the exhibits are big physical constructions you can climb on or fiddle with. Highlights were the raft lying on solids of constant width, and the tricycles with square wheels rolling on a catenary track.

(Behind me in that photo is a pretty cool exhibit which lets you tile surfaces of different curvatures with regular polygons by holding them on with a vacuum. When the vacuum turns off it all falls apart!)

There was a cool gizmo which presents you with a polyhedron on a screen and lets you extrude or squash or twist it to create a funky new shape, using an enormous trackball that wouldn’t look out of place in a cheap 90s fantasy film. There was a cabinet on the back of the machine showing off some shapes that had been 3d printed.

We spent a little while at a table which had a bucket of itsphun plastic polygons, sliding them together to create some shapes. Following the list of ingredients on a help sheet, Helen set to work trying to construct a dodecahedron, adding the requirement that no two touching faces have the same colour. She didn’t manage it, and I didn’t think it could be done, but I’ve just googled it and George Hart proves her right.

The basement had a few more exhibits, quite a few of which were unfortunately broken – since so many of the exhibits rely on technological wizardry, they have trouble standing up to the beatings tiny maths-mad people can inflict. A maintenance chap was valiantly trying to glue a bit of an exhibit back on while kids were crawling over it.

Hidden away at the back was a very sturdy American encryption machine, which was very satisfying to use.

There were some touchscreen information displays dotted about, promising to explain the exhibits in a bit more depth, once you’ve played with them and had time to think and discover for yourself. I would’ve preferred something more direct, though.

We just had a few minutes to look at the shop before the museum closed. That was lucky, because we could have spent a fortune in there! I don’t think I’ve ever seen so much fun maths stuff gathered in such a small space. It was all very reasonably priced, too. We bought a small pack of itsphun shapes – enough to make each of the Platonic solids – for a couple of dollars, a t-shirt featuring the excellent MoMath logo, a teeny tiny card game called Iota, which is like a cross between Set and Scrabble, and a set of Platonic solid dice.

All in all, we were only in the museum for about an hour. Helen really liked it, despite being very much disinclined to like maths. I enjoyed it, but nothing really blew me away. It’s worth popping in if you’re in New York. I think the real value might be in the events they run, with the very best invited guest speakers showing off all sorts of interesting maths.

**More information: **momath.org

One of the many jobs we’re gradually getting round to in our new flat is that of tiling a small section of the kitchen surface, which for some reason was left blank by the original builders and all intervening owners. And what better thing to tile it with than binary numbers?

The section of surface in question sits at the right hand end of the counter, which for mysterious reasons doesn’t go all the way to the wall, and a roughly 10cm wide strip of bare wood, lower than the surface, was just asking to be tiled. Mosaic tiles, available from all good DIY shops, can be arranged in any pattern you like, and taking our inspiration from the fact that it would just about fit five 1.5cm-wide tiles across with appropriate spacing, we decided to tile it using black and white tiles to spell out a message in binary. This would suit our existing colour scheme, and be both mathematical and subtle – in that it will pretty much look random to anyone not fluent in ASCII.

The strip of surface in question was long enough from front to back that we could fit exactly 35 rows of tiles in the other direction, which gave us our message length – each letter of the alphabet could be converted to a number between 1 and 26, which can adequately be expressed using one row of five tiles in binary, meaning we had to come up with a message 35 characters long.

After a lot of discussion, online surveys and soul-searching, we settled on a geeky reference appropriate to the kitchen items usually stored in that area, which is our set of spice racks. From the Dune series of novels by Frank Herbert, the phrase “WHO CONTROLS SPICE CONTROLS THE UNIVERSE” fits perfectly, is not too personally revealing so would be fine if future owners of the property figure it out, and most importantly, looks basically random when spelt out in binary.

Our testing of phrases for this last property was accomplished using a flashy MS Excel spreadsheet, which can be downloaded if you would like one, and thanks to some exceptionally hacky formulae and conditional formatting, will automatically convert a message typed into the box at the top into a pattern of black and white squares for you to play with. This could also be accomplished using e.g. Python, if you’re a proper person, but I use spreadsheets for everything and I don’t care what you think.

The tiling itself was relatively straightforward – we bought a set of black and white square tiles which were already attached to a sheet of webbing, designed to be laid in a checkerboard pattern. We promptly ripped them all off the backing in order to rearrange them into lovely binary. With 3mm spacers in between, the tiles fit snugly in the space, and after some helpful advice from my dad concerning the best type of spreader to use, we soon got them all stuck down in nice straight lines.

The finished tiled surface is (pretty much) level and almost looks like we got a real person in to do it. It also, and this is my favourite part, looks to the untrained eye like any old randomly tiled kitchen counter. The same technique could probably be applied to bathroom walls, splashbacks, the tops of small tables, or anything that usually gets tiled and where it’s not too fiddly to use mosaic tiles.

Here’s a photo gallery of the tiles going on, and the finished product with spice racks back in place. Enjoy!

Click to view slideshow. ]]>Snowflake Seashell Star is a new mathematical colouring book, by Alex Bellos and Edmund Harriss, aimed at the lucrative ‘grown-up colouring books’ market that’s sprung up recently, heavily intersected with people who are interested in maths – the book can be used as a regular colouring book, but contains lots of interesting mathematical things, and mathematicians will love it. I wouldn’t have expected anything less from maths adventurer Bellos and mathematical artist and tiling fan Harriss, whose personalities both come through in the book – from the beautiful illustration to the playful style (and there’s a sneaky Harriss Spiral in there too).

The first thing I did in order to properly review the book was check an important mathematical fact, in case anyone was worried. And yes, **everything in it is colourable using four colours or fewer**. Phew.

In fact, everything in it is gorgeous and pretty and, along with some of the regulars at Manchester MathsJam, we looked through it and recognised almost every page as either an existing beautiful mathematical image, or some visual representation of a mathematical idea. We got stuck on identifying images for a minute, but luckily the book also includes a section at the back which explains what each page shows, and why it’s cool. This is also the location of my top favourite pun of the book, “Petit Fouriers” which is the name of the page featuring some gorgeous patterns made by applying Fourier transforms to simple shapes.

The range of types of maths on display is super, from a 7-way Venn diagram to hyperbolic projections, fractals, aperiodic tilings and prime numbers. Without spoiling too much of what’s in there, the book is split into two sections – the first is things you can colour in, and the second half includes some partly-finished things which you finish off using mathematical principles and occasionally a little colour-by-numbers.

Aside from being a beautiful object, the maths is all solid, and it was pointed out to me that this would be the ideal book to bridge the gap between someone who considers themselves a maths person and an arty person (let’s not start an argument about whether these two are distinct things, but some people like to think of themselves as one or the other) – with a good opportunity to explain some nice interesting maths concepts, expressed in a visual way, and also a good chance to develop aesthetic sensibility and colour a thing in pretty.

This is before you even get to the thing where colouring in is now being used as stress relief and for relaxation by a lot of people (something something mindfulness), which I imagine this book with its beautiful images would be great for. Whether you’re methodical or artistic, you can make use of it as a starting point.

The book is out today, and available from all good bookshops. You’ll have to get your own colouring pencils though. Mine are busy.

Snowflake Seashell Star at publisher Canongate’s website.

]]>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.

]]>Some mathematicians at Harvard have formed an organisation called “Gender inclusivity in mathematics”, which is dedicated to “creating a community of mathematicians particularly welcoming to women interested in math and reducing the gender gap in Harvard’s math department.”

They’re running a series of talks by invited speakers and discussion events, and hope to run a conference on women in mathematics this academic year.

**More information: **Gender inclusivity in mathematics at Harvard

*via Nalini Joshi on Twitter*

The National Numeracy campaign, a British charity aiming to improve numeracy in adults as well as children, has relaunched its Parent Toolkit as the Family Maths Toolkit. It contains advice for parents on promoting good attitudes towards maths, as well as ideas for activities to get the whole family enthused and practised in maths. There’s also a section with information to help school teachers support parental engagement with maths.

**Visit: **Family Maths Toolkit

Khan Academy has partnered with Pixar to produce a subsection of their site which explains some of the maths involved in computer animation. There are the usual videos explaining individual topics, but also plenty of interactive diagrams so you can play along at home. It’s also nice to see some “Get to know …” videos, which present real animators who work at Pixar, talking about why they got into computer animation.

**Visit: **Pixar in a Box, at Khan Academy

Here’s his abstract to the paper, which is titled “The Erdős discrepancy problem”.

We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erdős. In fact the argument also applies to sequences $f$ taking values in the unit sphere of a real or complex Hilbert space.

The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when $f$ is replaced by a (stochastic) completely multiplicative function ${\bf g}$. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when ${\bf g}$ usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.

As Terence mentioned in the abstract, his proof builds on the work of the collaborative Polymath5 project, which took place on Timothy Gowers’s blog.

We last talked about this problem in February 2014, when Boris Konev and Alexei Lisitsa made the news with an attack through the SAT problem (that post contains a good video by James Grime explaining the problem, by the way). Tao refers to this work in an example, which seems to have guided his thinking about the proof.

The comments field on Tao’s arXiv submission mentions that he’s submitted it to the newly-created arXiv overlay journal *Discrete Analysis*, which was only announced last week. What a coup for open access mathematics!

The Erdős discrepancy problem by Terence Tao, on the arXiv

The Erdos discrepancy problem via the Elliott conjecture on Tao’s blog a week ago, which seems to have been the big breakthrough.

Explanation and notes on the Erdős discrepancy problem, on the Polymath wiki.

]]>The main Breakthrough Prize has been awarded annually since 2012 for outstanding achievements in life science and physics, and since 2014 also mathematics. The awards, founded by a collection of extremely rich science bods including Mark Zuckerberg and Sergey Brin among others, are accompanied by a lavish televised ceremony with guests from technology, entertainment, business and academia, and presented to the scientists by Hollywood stars. Winners, of which there are several in each category, are awarded a prize of \$3 million each, and give public lectures about their work throughout the rest of the year.

The junior prize, which will be judged by a combination of peer review with the other contestants, then evaluation by selection committees and a final judging panel, consists of a \$250,000 post-secondary scholarship for the winning student.

Entrants have to be aged between 13 and 18, and submissions must be in English but can be from anywhere in the world. The closing date for entries is 7th October 2015. The competition is being jointly run by Khan Academy.

This journal was published by the Maths, Stats and OR Network 2001-12, then by the Higher Education Academy in 2013. The first new issue for two years, published by a volunteer group coordinated and supported by **sigma** and the Greenwich Maths Centre, is volume 14 issue 1.

This issue includes articles about maths support, active learning of game theory, support for numerical reasoning tests in graduate recruitment, an implementation of the Maths Arcade, and an article about the new maths learning space at Sheffield Hallam University at which I am going to work later this month, written by my new head of department (you can just about see my office-door-to-be in figure 2).

Submissions are encouraged, which could be case studies, opinion pieces, research articles, student-authored or co-authored articles, resource reviews (technology, books, etc.), short update (project, policy, etc.) or workshop reports and should be of interest to those involved in the learning, teaching, assessment and support of mathematics, statistics and operational research in higher education.

]]>