Plus Blog

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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December 15, 2014

We have a Mega Menger! Thanks to the tireless effort of students, the Centre for Mathematical Sciences in Cambridge (home of Plus) now has a piece of the largest distributed fractal model in the world. It's a model of a Menger sponge made out of over 48,000 business cards that are held together without any glue or sticky tape. Sets of six cards were folded up to form little cubes, a total of 8,000 of them, which together form an approximation of the sponge. The final result, which now adorns our common room, weighs over 90kg and is 1.5 metres tall.

Menger sponge

A model of the Menger sponge at the Centre for Mathematical Sciences in Cambridge.

Menger sponge

Little cubes made out of six cards.

The sponge model build was started off at an public event back in October, using thousands of little cubes that had been pre-built by students from twenty Cambridgeshire schools, coordinated by the Further Mathematics Support Programme. Enthusiastic members of the public and university students then finished it off, at one point working by the light of a single bulb when the building had shut down for the night. The effort is part of the MegaMenger project, which aims to build fractal models in multiple sites worldwide. If you would like to see our Mega Menger model, come to the Maths Faculty event at the 2015 Cambridge Science Festival, Saturday 21 March 2015, 12 noon — 4pm. You can find out more about fractals and the Menger sponge here.


Menger sponge

Students building the sponge by the light of a single bulb.

Menger sponge

Work in progress.

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