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September 4, 2013

What is the shape of the Universe? Is it finite or infinite? Does it have an edge?

In their new show X&Y Marcus du Sautoy and Victoria Gould use mathematics and the theatre to navigate the known and unknown reaches of our world.

Through a series of surreal episodes, X and Y, trapped in a Universe they don't understand and confronted for the first time with another human being, tackle some of the biggest philosophical and scientific questions on the books: where did the Universe come from, does time have an end, is there something on the other side, do we have free will, can we ever prove anything about our Universe for sure or is there always room for another surprise?

Marcus and Victoria met while working on A disappearing number, Complicite's multi award-winning play about mathematics. X&Y has developed from that collaboration and pursues many of the questions at the heart of A disappearing number.

X&Y is on at the Science Museum in London 10 - 16 October 2013. Click here to book tickets.

You can read about A disappearing number, an interview with Victoria Gould and several articles by Marcus du Sautoy on Plus.

August 29, 2013
paper map

If you like the Rubik's cube then you might love the Magic Cube. Rather than having colours on the little square faces it has number on it. So your task is not only to put the large faces together in the right way, but also to figure out what this right way is. Which numbers should occur together on the same face and in what order? Jonathan Kinlay, the inventor of the Magic Cube, has estimated that there are 140 x 1021 different configurations of the Magic Cube. That's 140 followed by 21 zeroes and 3000 more configurations than on an ordinary Rubik's cube.

To celebrate the launch of the Magic Cube, Kinlay's company Innovation Factory is running a competition to see who can solve the cube first. To start it off they will be shipping a version the puzzle directly to 100 of the world's leading quantitative experts, a list that includes people at MIT, Microsoft and Goldman Sachs.

You can join too by nominating yourself (or someone else). Innovation Factory will accept up to 20 nominees (in addition to those that have already been picked). The competition will launch in September and run for 60 days. To nominate someone please send an email to MagicCubeCompetition@IF-Chicago.com, giving the name and email, mailing address of the nominee and a brief explanation of why you think they should be included in the competition. If you don't get accepted, don't worry — the Magic Cube will go on sale after the competition has ended.

The winner will receive lots of glory and a metal version of the Magic Cube precision-machined from solid aluminium, and they will be featured on the Innovation Factory website.

As a warm-up you can read about the ordinary Rubik's cube on Plus.

Good luck!

August 16, 2013

The arXiv is an electronic repository containing scientific papers, both published and as pre-prints, in the areas of physics, mathematics, computer science, mathematical biology, quantitative finance and statistics. That's a lot of areas and there are a lot of papers on the arXiv: over 860,000. But physicists Damien George and Rob Knegjens have got to grips with them all. They have constructed a clever algorithm to visualise the vast paper galaxy. As explained on their website, the algorithm is based on two "forces" that determine the papers' positions on a map based on citations between papers: "each paper is repelled from all other papers using an anti-gravity inverse-distance force, and each paper is attracted to all of its references using a spring modelled by Hooke's law." Each paper on the map is represented by a circle and the area of that circle is proportional to the number of citations the paper has.

The result is not only pretty but also informative and interactive, showing how papers clump together and how different areas relate. Click here to use the map and to find out more.

paper map

Click on the image to use the map.

August 9, 2013
Earth

A Kawasaki new rose. Image: Gila Oren.

If you've enjoyed the origami articles on Plus and live in or near London, then here's your chance to see some origami live. The Japan Foundation has invited Toshikazu Kawasaki, creator of the Kawasaki rose, to give a special talk on 27 August 2013. Kawasaki, a renowned origami theorist and maths teacher by profession, will talk about his approach to the paper craft with actual origami demonstrations. His talk will also explore his role as an origami master and how his passion for origami and maths led him to become the world's first doctor of origami. During the talk the audience can also put their origami skills to the test with the instruction of Kawasaki.

The event will take place at The Japan Foundation, Russell Square House, 10-12 Russell Square, London WC1B 5EH. It's free but booking is essential. To reserve a place, please email your name and the title of the event to event@jpf.org.uk.

August 1, 2013

This year's British Science Festival will take place from 7 - 12 September at the University of Newcastle and there's plenty of (mostly free) maths on offer.

You can take part in some pioneering maths at PrimeGrid: Search for a world record prime number on 7 September. Learn about prime numbers with more than a thousand digits, the history of the search for large primes, and how you can use your computer to join PrimeGrid and find new primes of your own.

The randomness show on 8 September explores randomness in your life, and the ways in which maths can order the chaos around us. You can delve into mathematical history, discover how maths can be used to predict the future and uncover the hidden maths inside you.

Be a maths millionaire on 11 September is all about famous mathematical equations and the adventures of the mathematicians working on them, including those that make Google work.

If you're interested in economics then An insight into financial markets and trading on 11 September may be for you: it will have a real time interactive workshop exploring financial markets. You can get to grips with data, charts and financial information in the trading room. This event costs £4.

Slide into the maths museum on 7 September explores the maths needed to crack the fastest slide and build skateboarding tracks. The UK's first museum entirely dedicated to mathematics wants to know what you would like at the museum. This event costs £10.85 for adults and there are various concession rates.

For the beady-eyed, see if you can spot Dr Math's giant logic puzzles all around the festival and how many of his mind-bending brain-twisters you can solve.

Space for some events is limited, so book now to reserve your place at www.britishsciencefestival.org or call 08456 807 207 for more information.

July 16, 2013

Number theory is famous for problems that everyone can understand and that are easy to express, but that are fiendishly difficult to prove. Here are some of our favourites.

The Goldbach conjecture

The Goldbach conjecture is named after the mathematician Christian Goldbach who formulated it in the middle of the eighteenth century. It states that any even natural number greater than 2 can be written as the sum of two prime numbers.

Leonhard Euler

Leonard Euler (1707-1783) corresponded with Christian Goldbach about the conjecture now named after the latter.

It is easy to see that this is true for the first few even numbers greater than 2:

4=2+2
6=3+3
8=3+5
10=5+5=3+7.

This seems so straightforward you might be tempted to try and prove it yourself — and you'd be in very good company as some of the brightest mathematical minds have been chiselling away at the conjecture ever since it was first pronounced. But so far without success. The closest result that has been proved, in 1995, says that every even number is the sum of at most six primes.

There is a similar statement, called the weak Goldbach conjecture, which says that every odd natural number greater than 5 is the sum of three primes. Again we can see that this is true for the first few odd numbers greater than 5:

7 = 3+2+2
11=3+3+5
13=3+5+5
17=5+5+7.

This statement is called "weak" because once someone finds a proof for the ordinary "strong" Goldbach conjecture, the weak one can be deduced from it.

In 1938 Nils Pipping showed that the (strong) Goldbach conjecture is true for even numbers up to and including 105. The latest result, established using a computer search, shows it is true for even numbers up to and including 4 x 1018 — that's a huge number, but for mathematicians it isn't good enough. Only a general proof will do.

Perfect numbers

A perfect number is a number that's the sum of all of its divisors (excluding itself). For example, 6 is a perfect number because its divisors (apart from 6 itself) are 1, 2 and 3, and

6 = 1 + 2 + 3.

The next perfect number is 28, which has divisors 1, 2, 4, 7 and 14, and:

28 = 1 + 2 + 4 + 7 + 14.

Leonhard Euler

Euclid, depicted with a compass in Raphael's painting The school of Athens.

The next three perfect numbers are 496, 8128 and 33,550,336.

The gaps between perfect numbers are as wide as their discovery has been painstaking. The first four perfect numbers seem to have been known to the Greeks, the fifth and sixth weren't written down explicitly until the 15th century and the seventh followed in the 16th century. Today we know of 48 perfect number, the largest of which has over 34 million digits. All of these 48 are even. This raises two questions:

  • Are there infinitely many perfect numbers?
  • Are there any odd perfect numbers?

So far no one has been able to answer these questions with a conclusive proof.

One thing that was already known to the Greek mathematician Euclid over 2,000 years ago is that if p is a prime number and 2p-1 is also a prime number, then 2p-1(2p-1) is an even perfect number. For example,

21(22-1) = 6
22(23-1) = 28.

The 18th century mathematician Leonhard Euler proved that every even perfect number is of this form. The largest known perfect number, the one with over 34 million digits, is

257885160 x (257885161−1).

This leads us straight to our next number mystery.

Mersenne primes

Marin Mersenne, (1588-1648).

Marin Mersenne, (1588-1648).

Prime numbers are those numbers that are divisible only by themselves and 1. The first few are 2, 3, 5, 7, and 11. Unlike for perfect numbers we do know that there are infinitely many of them. The proof for that was furnished by Euclid, however there is no easy recipe that generates all the prime numbers. This is where numbers of the form 2p-1, where p is a prime, come in useful. These are called Mersenne numbers, after the French monk Marin Mersenne (1588-1648) who studied them, and they have a good chance of being prime themselves.

The question is, are there infinitely many such Mersenne primes? Mathematicians believe that there probably are, but again nobody has as yet been able to prove that conjecture. A total of 48 Mersenne primes have been found so far, the largest, discovered in January 2013, being

257885161 − 1.

These correspond to the 48 known perfect numbers. The search for larger and larger Mersenne primes continues, as does the search for a conclusive proof that there are infinitely many.

And there is more...

Another favourite number theory mystery is the twin prime conjecture, which states that there are infinitely many pairs of primes that are 2 apart. There's been recent progress on this, so we refer you to our news story. One mystery that has been solved, after over 350 years of effort, is Fermat's last theorem. We recently celebrated the twentieth anniversary of the announcement of its proof — you can find out more here. That's probably enough to fill a minute, but if you haven't had enough you can read more about number theory, prime numbers, Mersenne primes and the search for larger and larger primes here on Plus.

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