Plus Blog

July 24, 2014

In the process of writing an article on curvature we got entirely distracted by making a geogebra worksheet showing the tangent, normal and osculating circle to any smooth function. The meaning and mathematics of all these terms is revealed in this article, but if you fancy getting your hands dirty yourself, have a play with the worksheet below. Please post your favourite function as a comment – the curvier and wigglier the better!

You can use this geogebra worksheet to see the tangent, normal and osculating circle of any smooth curve you choose - just change the equation f(x) in the left-hand panel.

July 18, 2014
The principles behind the Forth Bridge

IMage courtesy of the Department of Civil and Environmental Engineering, Imperial College London.

This is one of friend of Plus Ahmer Wadee's favourites images from the book 50 visions of mathematics. It is of a demonstration at Imperial College in 1887 of the mathematical principles behind (or should that be underneath?) the Forth Bridge. The bridge was the largest spanning bridge in the world at the time and the technique behind it was an innovation, essentially balancing the forces involved using cantilevers. The men on the chairs (Sir John Fowler and Benjamin Baker) represent the piers of the bridge and the load on the bridge, in this case Kaichi Watanabe, one of the first Japanese engineers to study in the UK, is supported by the tension (in the men's arms and in the ropes to the anchors) and compression in the structure.

So, what holds up the Forth Bridge? Why, maths of course!

You can read more about the maths of engineering in Constructing our lives on Plus

July 16, 2014

If you love physics and making movies then this is for you. The Foundational Questions Institute (FQXi) is excited to present Show me the physics!, its first-ever video contest. Anyone can submit a video conveying the joys of physics to win a top prize of $10,000, and there are very attractive runner-up prizes too.

Whether you're a physicist or just a physics geek, and whether it's the geometry of space time (see top video on the right) or quantum immortality (see bottom video), all sorts of submissions are welcome. The aim is to enthuse non-physicists and provide a creative and visual space for the discussion and exchange of ideas. Examples of suitable topics are:

  • Unsolved physics mysteries
  • Physics experiments being carried out
  • Tales of physics discoveries
  • Accounts of how physics has improved our lives
  • Physicists, inventors, teachers, and others talking about their passion for physics
  • Fictional stories in which real physics plays a central role.

The closing date is August 8, 2014. See here here for rules and submission guidelines and here to see the current entries.

Good luck!

FQXi are our partners on the Information about information and Science fiction, science fact projects.

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!

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