Plus Blog

December 19, 2014
It's a Gömböc!

Still looking for a special present for a special person? Then what about a Gömböc: this strange egg-like shape wriggles around with an apparent will of its own and, even better, barely exists at all.

A Gömböc is a 3D convex shape with one stable and one unstable equilibrium point. The reason it wriggles around when you put it down is that it's "trying to" rest on its stable equilibrium point, and since there's only one of them it takes a little dance to find it. for other objects this doesn't happen because they have more stable equilibrium points, so all they need to do to find one when you put them down in an unstable position is fall over. If you change a Gömböc even only a tiny bit, you'll create extra equilibrium points or stop it from being convex, so you'll stop it from being a Gömböc. This is why Gömböcs teeter on the brink of existence — the smallest change and they're gone.

For a very long time people believed that Gömböcs didn't exist, but recently the Hungarian mathematicians Gábor Domokos and Péter Várkonyi proved that they did (find out more here). You can purchase your very own Gömböc on the Gömböc website (though be warned, it's a little pricey!).

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December 18, 2014

If you're feeling a bit weary in the run up to Christmas (I know we are!) we thought you might like to put your feet up and have a cup of tea instead of whatever it is you are working on. You can relax because every thing you've ever produced or will produce is already encoded in a number known as Champernowne's constant, consisting of every positive whole number listed after the decimal point:

0.1234567891011121314151617….

This is because this number is normal, which implies it contains a copy of every finite string of numbers in its infinite decimal expansion. This includes every article we have ever written or will write, every song anyone's composed, every film shot, every report written and every spreadsheet created, translated into numerical form (as it would be when stored on a computer). Lots of more famous numbers, such as $\pi $, e and $\sqrt 2$ are also thought to be normal. In fact although mathematicians know almost every number is normal, they have only been able to prove this is true for a handful.

Unfortunately for you, and for us as we try to finish a few more articles before Christmas, even though all the work we will ever write is already present in the digits of Champernowne's number, we aren't going to be able to head to the pub just yet. Our articles and everyone else's work are swamped by every other possible string of numbers and we're going to have to produce them the hard way after all.

hexagons

The idea is beautifully explored by the Argentine author Jorge Juis Borges in his short story "The Library of Babel" (published in the anthology Fiocciones). His library contains every book that is possible to write in a given alphabet, shelved in a seemingly endless complex of connected identical rooms (it looks a bit like the strange book filled place Matthew McConaughey found himself in towards the end of the Interstellar). The librarians were initially overjoyed to discover that the library contained every possible book. But their joy soon turned to despair when they realised that virtually all the books in the library would be nonsensical, the pages randomly filled with letters. They would spend their lives journeying through the endless identical rooms in a quest to find meaning among the books, knowing they were almost certain to every find it.

Borges' writings are full of such inventive and poetic explorations of philosophical and moral ideas. We came across this beautiful story when writing about normality and randomness for our book Numericon, and ended up devouring the rest of Fiocciones and have since gone on to read many others. You can also read more about the maths behind this story in The amazing librarian on Plus.

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December 17, 2014

Our image of the week is made of 2000 line segments!

Image by Hamid Naderi Yeganeh.

The image was created by Hamid Naderi Yeganeh running programs on a Linux operating system. Letting $n$ run from $1$ to $2000,$ the end points (given as coordinates in the plane) of each line segment are

  \[ \left(\left(\sin {\left(\frac{12\pi n}{2000}\right)}\right)^3, \left(\cos {\left(\frac{10\pi n}{2000}\right)}\right)^3\right) \]    

and

  \[ \left(\left(\sin {\left(\frac{8\pi n}{2000}\right)}\right)^3, \left(\cos {\left(\frac{6\pi n}{2000}\right)}\right)^3\right) \]    

You can see more of Hamid's images on this website and on the American Mathematical Society website.

You can see previous images of the week here.

December 17, 2014

No collection of popular maths books would be complete without a work by one of our very favourite authors, Ian Stewart. One we particularly like is 17 equations that changed the world. As the name suggests, it's about equations that have had a profound impact on humankind, from Pythagoras's theorem about right-angled triangles to the Black-Scholes equation about financial derivatives. It's a fascinating and accessible tour through some very interesting maths.

Stewart has written plenty of popular maths books, which are all well worth looking at, but there's another one we'd like to mention: Does god play dice is a great introduction to mathematical chaos, which inspired one of us to go and do a PhD in maths — she's never looked back!

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December 16, 2014
Travelling Salesman poster

The travelling salesman is one of our favourite mathematical movies. It's a tense thriller revolving around one of the most difficult open problems in maths, the P vs NP problem, and its potential to deliver the key to the world's most secret messages. Most of the movie is set in a single room, a secret government location, where four mathematicians are being debriefed as their highly classified project has been completed. An unexpected by-product of their work is a method for cracking the codes used to encrypt classified messages, giving rise to an intense debate between the mathematicians. Will their work be used for evil, by governments (or worse) to spy on all our communications and data? Would keeping it secret hamper medical advances and scientific discoveries that could be a force for good? The result is an intelligent movie full of suspense that takes maths, as well as mathematicians, seriously. Visit the movie's website for download or instant streaming.

To read about other mathematically inspired movies, read Maths, madness and movies.

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December 15, 2014

How are you feeling about Christmas lunch with the relatives? Worried you'll regress to your teenage years and all start pushing each other's buttons? The answer to this, as to all other questions, comes from maths: it pays to be nice!

Yes, yes it does.

For the full answer to this question, as well as a great way to while away the time between courses, get yourself or your loved ones a copy of SuperCooperators: Evolution, altruism and human behaviour or Why we need each other to succeed by Martin Nowak and Roger Highfield.

We were lucky enough to travel to Cambridge, MA, to meet Nowak and his colleagues and find out more about his research. You can read our interview with him and other stories about the maths of altruism in our package and ebook.

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