Plus Blog

July 11, 2014
Germany's 7-1 victory over Brazil in fractal form

Image by Lasse Rempe-Gillen – click on the image to see a larger version.

To mark Germany's historic win over Brazil in the World Cup semifinal this week, Lasse Rempe-Gillen (from the University of Liverpool) created this beautiful image. It shows the behaviour of a model that describes the phenomenon of phase-locking, something that can be seen in the synchronising flashes of fireflies or when a roaring stadium of football supporters gradually clap or stamp in unison. The image is related to recent research and you can read more in our news story Maths, metronomes and fireflies.

The grey parts of the image show where the model behaves chaotically – here even small changes in where you start can cause drastically different results in the model. The coloured parts of the image show where the model behaves in a more regular fashion where small differences won't dramatically change the results. This is because the model has attractors, special sets of conditions that create similar behaviour, either settling on a single outcome (called a fixed point) or running through a predictable cycle of outcomes. And in honour of the historic 7-1 score from the match, Rempe-Gillen's image has attractors of period 7 (with a repeating cycle of 7 points) and period 1 (a fixed point).

In contrast his image below has no periodic attractors, symbolising the other, goalless, semifinal between Argentina and Holland.

Holland v Argentina's 0-0 semifinal in fractal form

Image by Lasse Rempe-Gillen – click on the image to see a larger version.

You can read more about chaos, fractals and football on Plus!

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!

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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.


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


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

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 appears 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.

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 favouritenumber.net or read Alex's new book Alex through the looking glass (reviewed in Plus).

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 appears 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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