Saturday, 7 June 2014


I was fascinated that one of the Secret Mathematicians discussed in Marcus du Sautoy's wonderful Gresham College / London Mathematical Society lecture a couple of weeks ago was Rudolf Laban.  Some years ago a friend of mine was studying dance theatre at the Laban Centre (now part of Trinity Laban Conservatoire). She was studying hard for her exam on Labanotation, which is Laban's method of notating dance.  I looked over her shoulder and was struck by the beautiful mathematics behind the notation.  (Apparently this wasn't the right thing to say to somebody who needed to pass an exam in  a subject with which she had no natural affinity - which I'm happy to say she did.)

Mathematicians tend to like good systems of notation, and notating something as fluid and instantaneous as dance is a huge challenge.  I'm not qualified to say how effective Labanotation is for recording dance - my friend and her fellow dance students certainly didn't find it intuitive.  But it has been used for one invaluable project. 

Alec Finlay has notated Archie Gemmill's goal for Scotland against Holland in the 1978 World Cup.  (You can buy the book from Amazon - I can't find a publishers' website.)  That the notation which drive my dancer friend to distraction has been used to record the greatest goal ever scored in a football match is a demonstration of the importance of notation (which I regard as a branch of mathematics).

Tuesday, 8 April 2014

109th Carnival of Mathematics

Welcome to the 109th Carnival of Mathematics!

Shibuya 109 department store, Tokyo, Japan

109 is a prime.  I'm pleased to say that it is a happy, polite and amenable number (for definitions see, for example,  It is a Chen prime - that is a prime number p such that p+2 is either prime or the product of two primes: in this case 111 = 3 times 37.  You may remember that from Carnival 107, since 107 was also a Chen prime.

109  is the 24th term of the Euclid-Mullin sequence, a curious sequence of which each term is the smallest prime dividing the product of all the previous terms plus 1.  It's motivated by Euclid's proof that there are infinitely many prime numbers.  It begins 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, ... and only the first 51 terms of the sequence are known, since the 52nd is the smallest prime factor of a 335-digit number of which the divisors haven't yet been found.

If we express 1/109 as a decimal, the last six digits of the recurring sequence are 853211, which gives us the first six Fibonacci numbers in reverse order.

Now to the Carnival.

It's always good to start with humour.  In the last few days, xkcd has been in excellent form: here is one on teachers' statistics.  And having spent much of my life writing mathematical modelling software, I particularly enjoyed PHD Comics on software testing which, as so often, hits the mark.

How about the amazing Domputer?  Watch the video of the domino computer at the Aperiodical.

What does "real-world" mean with respect to maths in the classroom?  dy/dan, mathalicious and Math goes Pop! have been debating this difficult question.

Maria Droujkova has claimed in an interview in Atlantic Monthly that "5-year-olds can do calculus": there is discussion at the De Morgan Forum.

GCHQ, the British Government Communications Headquarters which employs many mathematicians, has been in the news recently.  Tom Leinster wrote an opinion piece for the April Newsletter of the London Mathematical Society entitled "Should Mathematicians Co-operate with GCHQ?", which can also be found (with discussion) at the n-Category Café.

Paul Taylor examines Rubik's Cube at the Aperiodical, examining the periods of small sequences of moves.

Jimi Cullen has been exploring model checking - a tool from computer science with applications beyond that discipline.

MathFrolic links humour with mathematics and magic in reflecting on a mind-reading trick.

Another puzzle - can you see a pattern in apparently random data - leads The Well-Tempered Spreadsheet to Melting Fractals.

The Renaissance Mathematicus tells us about a dubious account of the origins of Newton's Principia in the American TV series Cosmos.

FlexMonkey has produced an app offering "a new type of calculator: numbers and operators appear as nodes in a network interface and the relationships between them define your calculations", which sounds intriguing.  And another calculator app, Tydlig, is reviewed by Aoife Hunt at the Aperiodical.

Max Tegmark's recent book Our Mathematical Universe, which argues that our universe is mathematics, has divided readers: some think it fascinating, others say its hypothesis is untestable and therefore scientifically meaningless.  For a typically entertaining discussion (which, if I understand correctly, goes along with both these views) see Scott Aaronson's blog Shtetl-Optimized on the topic.

(Aaronson's post has the the wonderful title "This review of Max Tegmark's book also occurs infinitely often in the decimal expansion of pi".  If you have a slow internet connection, feel free to save time by, instead of following all the links in this edition of the Carnival, simply going through the digits of pi till you find the content.  You might find some other interesting stuff on the way: if so, submit it to the next Carnival!)

From deep or possibly meaningless ideas about whether the universe is simply mathematics to a simple question about multiplication: what exactly do we mean by "6 x 4"?  Here are thoughts from reflectivemaths and flyingcolours.

For those looking for serious maths, Terry Tao gives a new proof of the Cayley-Salmon Theorem.  And of course Theorem of the Day features a different result every day - as I type this, today's theorem is Wagner's Theorem about planar graphs.

The wonderful lecture archive at Gresham College (disclosure: I'm a visiting professor there) contains the lectures by Raymond Flood, the Gresham Professor of Geometry: the video of the most recent lecture, on "Modelling the Spread of Infectious Diseases" should be available very soon.  And don't miss Colin Wright's wonderful recent lecture on "Notations, Patterns and New Discoveries": the title in my opinion seriously undersells its entertainment value!

And finally, the game at which everyone seems to be spending all their time this month - 2048.  (I also suggest you have a look at an old one - Lunar Lockout).  If you've discovered 2048, you'll appreciate this from xkcd.

If you have suggestions for Carnival 110, which will be hosted by Flying Colours Maths, please submit them here.

Sunday, 23 March 2014

A thought about Buffon's Needle

I am delighted that this blog will be hosting the April 2014 Carnival of Mathematics.  To find out more, see previous Carnivals, or suggest an article for inclusion, go to the Carnival of Mathematics page at the Aperiodical.  The Carnival will appear here soon after the deadline for submissions, which is 5 April.
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In my recent talk at Gresham College, I demonstrated a computer simulation of Buffon's Needle - a Monte Carlo method of finding an approximation to π.  The idea is that if one tosses a needle of length l onto a floor made up of planks of width t (with l < t for simplicity), then the probability that the needle crosses a junction of two planks is 2l/.  So if we toss n coins and m of them cross the junctions, then 2ln / tm will give us an approximate value for π.  By choosing suitable values of l and t, or n, and by being slightly lucky or by not stopping until you have the answer you want, you can get the approximation 355/113 by this method (in my lecture I got this value by tossing only two needles!), which demonstrates the power of the method if you know the answer in advance and use that information to full advantage in conducting the experiment.

If I choose numbers which are less designed to give me the answer I want. I find I can get π to reasonable accuracy - I just simulated 10,000,000 needles and got a result around 3.1488.  Since one is essentially sampling, statistical theory can give estimates for the likely proportion of tosses that will cross a line and hence for the accuracy I can expect.  But, it occurred to me, why bother tossing ten million computer-simulated needles?  Why not just calculate the expected value?

There's a good reason.  Suppose the probability that a random needle crosses a junction is p.  (In my lecture, where I chose l to be 710 and t to be 903 - notice the relationships to 355 and 113! - I have p almost exactly 1/2.)  Then the probability of m "hits" out of n tosses can be calculated by the binomial formula.  In particular, the probability that m is zero is (1-p)^n.  So for ten million tosses, I have a (finite) probability of (1-p)^10,000,000 that no needle crosses a line.  In that case, 2ln/tm is infinite. If at least one needle crosses a line, then the value of 2ln / tm is finite.  So when I calculated the expected value of 2ln / tm from ten million tosses, the result is infinite and the expected value of my approximation to π is infinity. Which is some way out.

Sunday, 2 March 2014

A prize which shows the diversity of applications of mathematics

A team from the Department of Mathematical Sciences at the University has just won the Guardian University Awards prize for Research Impact.  The team, led by Professor Ed Galea, is from the Fire Safety Engineering Group, and the official press release can be found here.

The work which has been honoured is concerned with signage: how can we make emergency signage more effective? The project involves dynamic signs, which can change depending on circumstances, so that if a potential escape route is blocked or unsafe, then operators can change the signs to divert people to safe routes.  This work is potentially life-saving for people escaping buildings in emergencies.

I'd like to make two general points about the value of mathematics.  First, its applications are extremely diverse: you might not have thought of mathematicians winning major prizes for working on signage!  But the mathematical algorithms underlying these active dynamic signs are quite literally going to make us all safer.  Secondly, mathematics cannot be done in isolation.  Work on projects like this involves collaboration with many other disciplines.  Computing, to implement the algorithms; engineering and architecture, to understand how buildings work; psychology, to understand how people behave in emergency situations and how they react to signs; and many others.

Mathematics really does make our lives better, especially when mathematicians work with others.

Wednesday, 1 January 2014

Five curious and interesting mathematical objects

Here (in no particular order) are five of the interesting mathematical objects I have gathered over the years. These ones come from three continents and are made of ceramic, wood, plastic and metal.

Number One - Trench's Triple Initial

Triple Initial

The late Kevin Holmes sold wooden puzzles at Covent Garden for many years.  These bespoke "Triple Initials" are solid cubes out of which three interlocking letters are carved.  

Number Two - Three hares with only three ears between them

Three hares tile

David Singmaster has traced the history of this motif - sometimes called the "Tinners' Rabbits", and in the UK particularly associated with Cornwall - to its origins in the far East, whence it spread along the Silk Road.  

Number Three - a strange map of the UK

Jigsaw map of UK

This wooden jigsaw map of Great Britain and Ireland has the remarkable property that every piece is in the form of characters from Alice in Wonderland.  A map of the UK based on a favourite British book - a wonderful present from my sister Rosie from her time in Jakarta!

Number Four: A non-snail snail ball

Non-snail snail ball

What is a "non-snail snail ball"?  Well, a snail ball is a cleverly constructed ball which rolls down a slope extremely slowly, contrary to our expectations.  I got one at Village Games in Camden Market many years ago. For an explanation, see this article by Stan Wagon.  The non-snail version looks and feels identical but doesn't have the "snail" property.  Mine came from (from whom I have lifted the picture).

Number Five - Dr. Nim

Dr Nim

Dr. Nim is an amazing marble-powered machine from 1966 which plays perfectly a version of Nim's game. This was a wonderful Christmas present from Noel-Ann Bradshaw.

Sunday, 17 November 2013

Highlights from MathsJam

It's already two weeks since the wonderful MathsJam conference: once again the most exciting maths event of the year.  I'm sorry that it has taken me so long to post about it.  There is so much I could write about from the talks and conversations - a number of unusual proofs of Pythagoras's Theorem, various interesting games, some nice problems, lots of ideas.

Here are just three outstanding memories:

  • Tarim's demonstration of the Dr Nim mechanical robot from the 1960s which plays a version of Nim's game perfectly.  This was particularly timely: Dr Nim will feature in my talk at the Science Museum Lates on the evening of 27 November.
  • Derek Cozens showing how to tie a tie one-handed.  i didn't manage to do it - but perhaps with some practice...
  • Pedro Freitas's demonstration of how to untangle a braid with a clever use of rational numbers
I intend to write a further post about Dr Nim when I have more time.

Sunday, 27 October 2013

Maths Education and Unintended Consequences

I should emphasise that this post expresses only my own personal views.

That policy decisions can have unintended consequences is well illustrated by events in maths education in England over the last fifteen years.

First, the introduction of Curriculum 2000 created a modular A-level syllabus in which AS-levels became stepping-stones on the way to the full A-level.  In principle I think this was a thoroughly good thing.  But it had a disastrous effect on mathematics in higher education.  With module exams a few months after students had moved from GCSE into A-level study, the gulf proved too great, AS-level results were spectacularly awful, the number of students taking A-level maths plummeted because potential candidates were discouraged by their results at AS-level, schools and colleges advised students not to study maths post-GCSE, applications to study maths at University dropped substantially, and University departments closed because they could not recruit enough students to be viable.  It took a decade for mathematics in higher education to recover.

There was also the unintended consequences of the GCSE data-handling coursework.  The country needs more statisticians, and it needs citizens with basic understanding of descriptive statistics.  So the introduction of a significant statistical assignment - the data-handling coursework - was largely welcomed by the statistics community.  It would give students more knowledge of this important subject and encourage more to study statistics at university.

But it didn't work out that way.  Students found the coursework time-consuming and tedious.  It put them off statistics!  Numbers taking statistics at university, whether as a subject in itself or as an option in business or science degrees, fell as a result.  University statistics departments have shrunk.  The GCSE data-handling coursework did a huge amount of damage to statistics.

As another example, consider the inclusion of mathematics GCSE in the government's schools league table data for five GCSEs at grade A-C.  The maths community was delighted when the old league table measure, just the proportion of candidates getting five A-C grades regardless of subject, was changed to require that the five GCSEs must contain Maths and English.  The feeling was that this would lead schools to put more effort into Maths GCSE.  But a consequence has been that many schools have been putting their students in for maths GCSE as early as year 9, and at every opportunity thereafter until they get the C pass, even if they have not covered most of the curriculum.    If a student can scrape enough marks for a C, they then drop the subject so that they can focus on the other GCSEs: so they may never cover a large part of the GCSE curriculum.  This is extremely damaging for students (and may have prevented many potential mathematicians from taking A-level maths).  Fortunately the government has now acted to prevent this abuse, but it is another example of unintended consequences.

So with this history of apparently desirable initiatives having adverse outcomes, I am nervous about the proposal for a new maths qualification for 16-18-year-olds.  Like many others, I believe the more maths people study, the better; I regret the English system which means that so few students study mathematics post-16; and I welcome the opportunity to allow post-16 study of mathematics for those who will benefit from a less intensive course than A-level (such as those who did not obtain A* or A at GCSE but who have the potential to gain from developing further mathematical understanding).

But there are dangers.  Where are the teachers going to come from?  Will resources move from A-level teaching to the new exam?  Worse, will students who would otherwise have taken A-level maths prefer the less intensive course?  Could the new mathematics exam lead to another drop in A-level numbers and impact on further study?  Are some of the potential top mathematicians of the future going to find themselves unable to study maths at university because they made a poor decision, or their school or college advised them badly, at 16?

The new maths exams should provide an opportunity for many to gain useful training in a subject that will benefit them throughout their lives.  But, once again, there is potential for unintended consequences which could damage maths education in England.