Plus Blog

October 1, 2013

If you walk around the two-dimensional plane you can keep walking indefinitely in all directions. You could say, in a very hand-wavy and intuitive sense, that there is infinity all around the edge of the plane, only of course you can never get to or see that edge. But still, you could try to imagine what happens if you shrink that infinity-edge to a point. Perhaps this would be a little like tightening the draw string on the rim of a fabric bag. Once you've tightened it, the bag is closed and resembles a deformed sphere.

Stereographic projection. Image: Jean-Christophe Benoist.

There is a way of making this intuition precise. Imagine a sphere and the plane that contains its equator. For any point p on that equatorial plane, draw the straight line that connects it to the North pole of the sphere. That straight line is going to intersect the sphere at some point. If p is on the exterior of the sphere it will intersect the Northern hemisphere of the sphere. If p is in the interior of the sphere the line will intersect the Southern hemisphere of the sphere. And if p lies on the sphere, so it's actually on the equator, it will itself be the intersection point. This way of associating to every point in the plane exactly one point on the sphere is called the stereographic projection.

It's easy to see that the further out on the plane your point p, the closer its projected image on the sphere is to the North pole. But no point on the plane projects to the North pole itself. The North pole is still available and as a sequence of points move out towards infinity on the plane, their projections move towards the North pole on the sphere. So you now declare that infinity is just a point (you draw the drawstring tight) and that its projection is the North pole of the sphere.

What you get is a continuous one-to-one correspondence between your plane together with infinity and the sphere. The two can be treated as one and the same thing. The plane with a point at infinity appended is called the Riemann sphere after the 18th century mathematician Bernhard Riemann (although strictly speaking the Riemann sphere is the complex plane with infinity appended — see here for more on complex numbers).

This is incredibly useful. You are probably familiar with functions that take the number line into itself. An example is $f(x)=1/x:$ it takes a number $x$ from the number line as input and returns $1/x$ as output. Unfortunately, the function is not defined at $x=0$ because division by $0$ is not allowed. However, as $x$ gets closer and closer to $0,$ $f(x)$ gets closer and closer to plus infinity if you're coming from the positive side, or minus infinity if you're coming from the negative side. If you could treat plus and minus infinity as one and the same ordinary point, then the function could be defined at $x=0$ and would be perfectly well behaved there. You can also define functions that take the plane into itself (the complex function $f(z)=1/z$ is an example) and again they may not be defined at every point because you have division by 0. However, by treating infinity as an extra point of the plane and looking at the whole thing as a sphere you may end up with a function that's perfectly tame and well behaved everywhere. A lot of complex analysis, the study of complex functions, is done on the Riemann sphere rather than the complex plane.
September 16, 2013
Iron filings

Do you remember those pretty field lines that emerge when you scatter iron filings around a magnet? In the case of a simple magnet the field is static; it doesn't change with time. But magnetism is just one aspect of something bigger: electromagnetism. You are at this very moment immersed in electromagnetic fields, generated by the Earth, the Sun, and even your toaster.

James Clerk Maxwell realised, in 1864, that electricity and magnetism were just two sides of the same coin and that light was made up of electromagnetic waves. He developed an elegant theory describing the unified force of electromagnetism and the equations that describe the dynamics of an electromagnetic field now carry his name.

Today's Guardian has a great introduction to Maxwell's equations, written by their science correspondent Alok Jha. And if you'd like to venture further into the wonderful world of field theory you can read our series of articles about what happened next, starting with Let me take you down, cos we're going to ... quantum fields.

September 12, 2013
butterfly

If you're looking for a change of scene next Tuesday lunchtime why not go along to hear Raymond Flood, Gresham Professor of Geometry, talk about Butterflies, Chaos and Fractals, 1pm on Tuesday, 17 September 2013, at the Museum of London. It's just one of a selection of great free public lectures given by Gresham Professors over the upcoming months.

Gresham College has been organising free public lectures for over 400 years, since the time of Elizabeth I. Gresham Professors, in subjects ranging from mathematics and law to divinity and rhetoric, give a range of lectures over the 3 years they hold the chair. Many of the greatest names in science and art have passed through Gresham's halls, including Christopher Wren and Robert Hooke.

We're also looking forward to hearing Carolin Crawford, Gresham Professor of Astronomy, talk about Quasars on Wednesday, 23 October 2013, at 1pm. And Flood's talk on Public Key Cryptography: Secrecy in Public on Tuesday, 22 October 2013 at 1pm, should be topical given the recent revelations of government surveillance.

Looking further ahead, the lecture User Error: Why it's not your fault on Monday, 20 January 2014 at 6pm from Tony Mann, the Gresham Professor of Computing Mathematics, is particularly comforting given that I spent yesterday trying to retrieve about a week's work that was lost somewhere in the bowels of my laptop!

To get you in the mood, why not read our article based on a previous Gresham lecture, Conic section hide and seek.

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.

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