Plus Blog

May 11, 2011
Triangulation of surface

Ah the humble triangle. This simple shape is one of the first we ever learn. But perhaps you didn't realise just how important triangles are...

A triangle is a three-sided polygon and comes in a variety of flavours. Some are to do with the length of the triangle's sides: equilateral — where all the sides (and all the angles) are the same size; isosceles — where two of the sides (and two of the angles) are the same size; and scalene — where none of the sides (or angles) are the same. The angles inside the triangle are also important. All the angles add to 180°. You can have acute triangles, where all the angles are less than 90° and obtuse triangles, where one of the angles is greater than 90°. And of course you can get right-angled triangles — one of the most important mathematical shapes inspiring Pythagoras' Theorem and trigonometry.

But triangles aren't just mathematically significant, they are also fundamental to the way we build in our environments, both physical and virtual. Triangles are special because they are exceptionally strong. Out of all the two-dimensional shapes we can make out of straight struts of metal, only a triangle is rigid. All other shapes can be deformed with a simple push if the shape is hinged at the corners (eg. a rectangle could be pushed over into a parallelogram). But not the trusty triangle, which explains its ubiquitous use in construction from pylons to bracing.

Triangles are also special because they are the simplest polygon — it is a common approach to a tricky geometrical problem, such as analysing a complex surface, to instead approximate it by a mesh of triangles. This approach is also used in the real world to achieve some of the exotic shapes we now see in modern architecture, such as the curved shape of 30 St Mary's Axe, aka the Gherkin, or the canopy over the courtyard in the British Museum.

This method of triangulation also is vital in building our virtual world. The fluid forms of the CGI characters we see in film and on TV are actually an incredibly fine mesh of triangles, in order that they can be stored and manipulated digitally.

Triangles — the simplest shape that makes our mathematical, physical and digital worlds go round.

You can read more about triangle strength and their role in our physical and virtual worlds on Plus. And you can find out all about their role in mathematics at Wolfram MathWorld.

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May 11, 2011

We came across the excellent Math Encounters Blog through their post Dimensional Analysis and Olympic Rowing, a response to the New Scientist article by Plus author, John Barrow.

Math Encounters is about those problems we might encounter any day where a little bit of maths can really help. Posts cover the maths of carpentry, drinking, management and much more. The maths can be quite surprising and is always clearly explained. And unlike many other popular science and mathematics blogs, it definitely doesn't shy away from the interesting details!

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April 20, 2011
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Is there maths in beach volleyball? Or show jumping? Or in Taekwondo? If there is, then Plus is going to find it.

But to know where to start, we need your help: we'd like to know which of the Olympic sports you'd most like to see covered in Plus. So please vote below — you can choose up to three sports. We'll do our best to cover your favourite sports in the run-up to London 2012 and our coverage will also be shared by our Olympic project Maths & sport: Countdown to the games.

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April 11, 2011

Topologists famously think that a doughnut is the same as a coffee cup because one can be deformed into the other without tearing or cutting. In other words, topology doesn't care about exact measurements of quantities like lengths, angles and areas. Instead, it looks only at the overall shape of an object, considering two objects to be the same as long as you can morph one into the other without breaking it. But how do you work with such a slippery concept?

One useful tool is what's called the fundamental group of a shape. Take the sphere as an example. Pick a point A on the sphere and consider all the loops through that point - i.e. you look at all the paths you can trace out on the sphere which start and end at your point A. You consider two loops to be equivalent if you can morph one into the other without cutting either of them. You can combine two loops p and q to get a third one by simply going around p first and then going around q. Also, if you traverse a loop in the clockwise direction, then this movement has an opposite, or an inverse, which is traversing it in the counterclockwise direction.

These two properties, that two loops can be combined to get a third and that every loop has an inverse (together with a couple of other properties), mean that the set of loops (where you consider two loops as equivalent if the can be morphed into one another) form a neat and self-contained structure called a group. It turns out that as long as your object is path-connected (there's a path linking any two points on it) this structure is the same no matter which point A you used as the base for your loops.

Now on the sphere every loop can be transformed into every other loop. In particular, every loop can be contracted into the trivial loop, which goes nowhere and is just your base point A. The fundamental group in this case is also trivial, in other words it contains just one loop. This is true not only for the perfectly round sphere, but also for a deflated football, and for any other 2D surface that's topologically the same as the sphere.

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But now think of the surface of a doughnut, also called a torus. In this case not all loops can be contracted to a point because they may wind around the hole of the torus and also around its body. A general loop may wind around the hole a total of m times and around the body a total of n times. It turns out that any two loops are equivalent if they wind around the hole the same number of times and also wind around the body the same number of times. The fundamental group of the torus is the same as a group structure you get from looking at ordered pairs of whole numbers (to be precise, it's same as the direct product ZxZ, where Z is the set of whole numbers). This is true not only for a perfectly round torus, but also for a really irregular and lumpy one. So the direct product of ZxZ, which is a well-understood structure, gives a good characterisation of tori, regardless of their exact geometry.

The concept of fundamental group is a powerful tool in topology, where you can't use precise measurements to describe an object. It's also connected to one of the trickiest problems of modern maths: the Poincaré conjecture. It seems obvious that any object with a trivial fundamental group is topologically the same as the sphere: a trivial fundamental group means that the object has no holes for the loops to wind around and if there are no holes, then the object can always be deformed into a perfect sphere. At the beginning of the twentieth century Henri Poincaré asked whether a similar statement was true for the 3D sphere (which is hard for us to visualise) and found the problem was a lot trickier. It took around 100 years to prove that the answer is yes.

Read more about the Poincaré conjecture, about topology in general and about groups on Plus.

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March 18, 2011
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Our cities are filled with buildings, roads, cars, buses, trains, bikes, parks and gardens. They are crisscrossed with power, water, sewage and transport systems. They are built by engineers, architects, planners, technologists, doctors, designers and artists. Our cities are shaped by our environment, our society and our culture. And each and every part is built on mathematics.

You can join Plus author and Charles Simonyi Professor for the Public Understanding of Science, Marcus du Sautoy, on a mathematical adventure in the city. Marcus and his team of mathemagicians are constructing walking tours of the city — but they need your help!

They are running a competition asking you to shine a mathematical spotlight on your city — it might be a piece of interesting architecture, a mathematical sculpture or the maths behind something more mundane, such as traffic lights — they want to hear from you! The competition opens on 4 April 2011 and closes 3 May 2011.

Winning entries will become part of a virtual mathscape of cities around the world and will help Marcus and his team develop their walking tours. And of course — you can win great prizes including subscriptions to Nature and becoming part of mathematical history by naming a mathematical object!

Hmmm... you could write about the maths behind architecture, engineering or traffic jams... there are so many possibilities! Where do you see the maths in your city?

For further information visit the Maths in the City site and follow them on Twitter and Facebook.

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March 15, 2011
The London velodrome

The London Velodrome.

With 500 days to go everyone here at the MMP is getting very excited about the 2012 Olympic and Paralympic Games. In July 2012 medals will be won, records broken and stories of triumph and tragedy will be told — and here at Plus we are looking forward to revealing the mathematics behind them.

The first of the 2012 Olympic venues has now been completed and in our latest news story, Leaning into 2012, you can find out the secret behind the shape of the track. And you can find out more about the maths behind the Olympics from Plus and the rest of the MMP at our new project Maths and Sport: Countdown to the Games.

"The Maths and Sport project will offer a new and deeper perspective for everyone about what is going on during the sporting events at the London Games," says John Barrow, Director of the MMP and author of our Outer space column. "Simple maths can be used to help show how Usain Bolt can run faster, find better ways to rig rowing eights or understand the statistics of scoring systems and drug testing."

So, whether you're an athlete on the track or just the armchair, it's time to start your training now!

You can read more about maths and sport in our Plus teacher package and find lots more activities and resources on Maths and Sport: Countdown to the Games.

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