Plus Blog

July 7, 2014

The Plus team cycling into the office… (Great video by St John's, Cambridge)

No theorems were solved at the Centre of Mathematical Sciences, home of Plus yesterday… Instead everyone was enjoying the Tour de France zooming through Cambridge! Whether it's finding the right gear, attacking the turns or building the perfect track, maths and cycling go together like lycra and shorts!

Gearing up for gold – how the invention of the chain drive and some simple ratios can give you speed on the flat and power up the hill.

Leaning into 2012 – why leaning into turns lets you go faster.

How the velodrome found its form – how maths helped create the iconic curves around this ribbon of track.

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July 4, 2014

Sine, cosine, and tangent — we do love our trigonometric functions! So imagine our bliss when we recently came across a function we had never even heard of before. It's called the haversine and it's defined in terms of the sine function:

  \[ \mbox{haversin}(\theta ) = \sin ^2(\theta /2). \]    
Great circle distance

The dotted yellow line is an arc of a great circle. It gives the shortest distance between the two yellow points. Image courtesy USGS.

The term "haversine" apparently comes from "half versed sine". To see why this function is useful, put yourself in the shoes of an intrepid traveller setting out on a sea voyage from Liverpool to New York. The first thing you'd want to know is how far you will have to travel. Ignoring islands, rocks, currents and other inconvenient factors, let's say that you will travel along the shortest path between the two cities. We know that the shortest path between two points is along a straight line, but that fact doesn't help you here. The straight line that connects Liverpool and New York cuts right through the Earth and you are not about to dig a tunnel.

You need the shortest path on the surface of the Earth, which is roughly spherical. On a sphere the shortest path between two points is along an arc of a great circle: that's a circle drawn on the surface of the sphere which is centred on the same point as the sphere and has the same radius. Any two points lie on a unique great circle, which they divide up into two arcs. The shortest path between the points is along the shorter of these two arcs.

So how do you calculate this great circle distance between two points $P$ and $Q$ on the Earth? First, recall that the locations of the two points are given by their latitudes, for which we will write $\phi _ P$ and $\phi _ Q,$ and their longitudes, which we will denote by $\lambda _ P$ and $\lambda _ Q.$ Write $R$ for the radius of the Earth, which is roughly $6,371$ km. The great circle distance $d$ between $P$ and $Q$ comes from the formula

  \begin{equation} \sin ^2{\left(\frac{d}{2R}\right)} = \sin ^2{\left(\frac{\phi _2-\phi _1}{2}\right)} + \cos {\phi _1}\cos {\phi _2} \sin ^2{\left(\frac{\lambda _2-\lambda _1}{2}\right)},\end{equation}   (1)

(where the angles are measured in radians).

Solving for $d$ gives

  \begin{equation} d = 2R \sin ^{-1}{\left(\sqrt {\sin ^2{\left(\frac{\phi _2-\phi _1}{2}\right)} + \cos {\phi _1}\cos {\phi _2} \sin ^2{\left(\frac{\lambda _2-\lambda _1}{2}\right)}}\right)}.\end{equation}   (2)

You’ll admit that this isn’t the simplest of formulae. If you were are a seafarer hundreds of years ago, armed only with sine and cosine tables to help you, working out the distance $d$ would prove pretty cumbersome. There’s a square root to take, as well as the inverse of the sine function .... argh!

But now let’s replace any expressions of the form $\sin ^2(\theta /2)$ by the haversine function. Expression (1) above becomes

  \[ \mbox{haversin}\left(\frac{d}{R}\right) = \mbox{haversin}\left(\phi _2-\phi _1\right) + \cos {\phi _1}\cos {\phi _2} \mbox{haversin}\left(\lambda _2-\lambda _1\right). \]    

The distance $d$ is now

  \[ d = R \; \;  \mbox{haversin}^{-1}\left(\mbox{haversin}\left(\phi _2-\phi _1\right) + \cos {\phi _1}\cos {\phi _2} \mbox{haversin}\left(\lambda _2-\lambda _1\right)\right). \]    

Working out the great circle distance between two points is so important in navigation that people in the old days produced tables giving the values of the haversine function and also of the inverse of the haversine function. This made seafarers’ lives a lot easier. Working out the distance $d$ only involved looking up two cosine values and two haversine values, adding and multiplying them in the correct way, looking up the inverse of the haversine function and multiplying by $R$— done!

The reason why the haversine function has come out of fashion is that with the help of calculators and computers it’s easy enough to work out the distance $d$ straight from formula (2). That’s why you don’t find a haversine button on your average calculator.

Let’s give it a go. Liverpool has latitude $53.4^\circ $ and longitude $-3^\circ $, and New York has latitude $40.71^\circ $ and longitude $-74^\circ $. These are measured in degrees. Converting them into radians (multiplying by $\pi /180$) gives $\phi _ L = 0.932^\circ $ and $\lambda _ L=-0.052^\circ $ for Liverpool, and $\phi _{NY} = 0.71^\circ $ and $\lambda _{NY}=-1.291^\circ $ for New York (rounded to three decimal places) Plugging these into expression (2), with the radius of the Earth $R = 6371$, gives a great circle distance of around 5313 km. Quite a way to go!

July 2, 2014
Brillouin zones

Image produced by R.R. Hogan, University of Cambridge.

This pretty picture looks like something you'd see through a kaleidoscope, but it's more than a bit of fun. It's an image of a two-dimensional crystal — but not as you would see it if you looked at it. Instead, it shows the Brillouin zones of the crystal, which give important information about how waves, such as light or X-rays, pass through it.

The picture, created by R.R. Hogan, is one of the images that appear in the book 50 visions of mathematics, which celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

To find out more about crystals and what can happen when you bombard them with waves, read Shattering crystal symmetries.

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July 1, 2014

Alex Bellos, one of our favourite maths authors, recently conducted a survey to find the world's favourite number. After polling more than 30,000 people from around the world, he found that the winner was ... 7!

We rather like 7 ourselves and decided that this is because 7 is the first prime number that really "feels" like a prime number, and because it's the most common result you get from throwing a couple of dice. Alex has come up with a different reason — see the video below. To find out more about the poll, see or read Alex's new book Alex through the looking glass (reviewed in Plus).

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June 25, 2014
Elephants in the Mandelbrot set

Image produced by Philip Dawid, using the program winCIG Chaos Image Generator developed by Thomas Hövel, © Darwin College, University of Cambridge, used with permission

This image appears to be a procession of elephants but is, in fact, a much-magnified small detail of one of the Mandelbrot set. The Mandelbrot set is a famous example of a fractal – mathematical objects whose structure is infinitely complex. Whether you're viewing them from afar or zooming in on them with a mathematical microscope, the same complexity is always visible. This self-similarity even extends to the some of the same structures repeating at all scales.

This unending complexity means that fractals live between dimensions. For example, there are shapes that are so crinkly, they are "more" than a one-dimensional curve, but not extensive enough to give a two-dimensional shape like a disc or square. Instead, they have a fractional, or fractal, dimension between 1 and 2, which is why Benoît Mandelbrot, the father of fractals, named them so.

Fractals famously appear in nature, from snowflakes to coastlines, and also have revolutionised mathematics by inspiring the field of chaos theory which is used in weather prediction and stock market analysis.

This beautiful image, created by Philip Dawid, is one of the images that appear in the book 50 visions of mathematics, that celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

You can read more about fractals and an article by Dawid on Plus.

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June 25, 2014
Simon Donaldson

Simon Donaldson. Image: Gert-Martin Greuel.

Five mathematicians turned into millionaires this week when they were awarded the Breakthrough Prize set up by Facebook founder Mark Zuckerberg and internet entrepreneur Yuri Milner. Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terry Tao and Richard Taylor will each receive £1.8m. That's more than the £930,000 that's awarded for the Nobel prize and should make up for the fact neither of the five is likely to ever to receive a Nobel: there isn't one for mathematics.

It's the first time this prize has been awarded to mathematicians. It joins a string of other prizes Milner has set up to reward physicists and life scientists. The idea is to give mathematicians (and other scientists) the "rock star status" they deserve. "We think scientists should be much better appreciated. They should be modern celebrities, alongside athletes and entertainers," Milner told the Guardian. "We want young people to get more excited. Maybe they will think of choosing a scientific path as opposed to other endeavours if we collectively celebrate them more."

Maxim Kontsevich

Maxim Kontsevich. Image: Gert-Martin Greuel.

The five mathematicians that have been chosen certainly made big splashes in the world of mathematics. You may have come across Richard Taylor and Terence Tao on Plus before: Taylor (among other things) helped Andrew Wiles solve a famous 350-year-old problem called Fermat's last theorem. Tao was one of the youngest people to ever receive the Fields medal (another prestigious maths prize) in 2006 and has delivered a range of ground breaking work, including on prime numbers.

Jacob Lurie

Jacob Lurie

The other three are equally eminent. Simon Donaldson (with whom we had the pleasure of working on a forthcoming series of articles) has provided deep insight into geometrical structures inspired by theoretical physics. Lurie, at 36 the youngest of the five, has also been honoured for work on the mathematical end of physics, as well as results that lie on the interface between geometry and topology on the one hand and algebra on the other. Kontsevich is also a theoretical physicist, but has been awarded the prize for an astounding range of contributions to pure maths.

Terence Tao

Terence Tao

Million dollar prizes are always a contentious issue. The money spent on a few individuals could be used to fund maths education at home or in developing countries, help support young researchers or those disadvantaged in some way, or fund important research projects. Milner and Zuckerberg's reply will no doubt be that a hefty prize tag, even if slightly unseemly, raises awareness and sends a clear message. Maths is essential, even the sort of pure maths that some of the laureates are being honoured for, which may at first sight seem useless. In a world of short attention spans money talks.

Richard Taylor

Richard Taylor.

UK mathematicians seem to have responded to the prize with mixed feelings. One unnamed mathematician cast some doubt on the prize's aim when he told the Guardian, "I find it interesting that they think it's possible to make rock stars out of people who do something that 99% of the population have no hope of understanding, and I include most professional mathematicians in that." This is something we don't agree with. While there is not much hope of explaining a mathematician's work to a general audience in great detail, it is certainly possible to give the gist of an area, and communicate its excitement, attraction and motivation. You won't convince everyone, just as you won't convince everyone of the worth of a piece of art, but you will always enthuse a few and interest many. If this is something mathematicians find hard to do themselves, then perhaps a little chunk of those millions should go towards funding maths communication.