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December 21, 2014

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

Scene from the play

Shane Shambhu as Ramanujan and David Annan as Hardy in the play A disappearing number. Photo: Tristram Kenton.

This beautiful sentence is from G.H. Hardy's 1940 essay A mathematician's apology. The work was Hardy's attempt to justify the pursuit of pure maths to non-mathematicians and to explain its motivation. It focuses on the beauty of maths and, unlike many other attempts to make maths appear attractive, takes pride in the un-applicability of pure maths — partly because something that has no applications can't do any harm. It's an understandable sentiment for a pacifist like Hardy at the time of WWII. And although Hardy was proved very wrong about the "purity" of his own field, number theory, which is today used in cryptography, it's still a fascinating and thought-provoking read.

In the Apology Hardy also mentions the mathematician Srinivasa Ramanujan, who played a defining part in Hardy's mathematical life:

"I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, 'Well, I have done one the thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.'"

Hardy's collaboration with the self-taught Indian genius was remarkable. It inspired the 2008 play, A disappearing number, which we explored in this Plus article. You can also listen to our podcast with actor and mathematician Victoria Gould reading a section from the foreword to Hardy's Apology.

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December 19, 2014

Srinivasa Ramanujan (1887 - 1920).

December 22nd would have been the 127th birthday of the legendary Indian mathematician Srinivasa Ramanujan. His story really is remarkable. Born in 1887 in a small village around 400km from Madras (now Chennai), Ramanujan developed a passion for maths very early on. By age 15 he routinely solved maths problems that went way beyond what his classmates were dealing with. He worked out his own method for solving quartic equations, for example, and even had a go at quintic ones (and failed of course, since the general quintic is unsolvable). But since he neglected all other subjects apart from maths, Ramanujan never got into university, and was forced to continue studying maths alone and in poverty. Only after a plea to an eminent mathematician, who described Ramanujan as "A short uncouth figure, stout, unshaven, not over clean," did Ramanujan eventually get a job as a clerk at the Madras Port Trust.

It was during his time at the Port Trust that Ramanujan decided to write a letter that was to change his life. It was addressed to the famous Cambridge number theorist G. H. Hardy who, accustomed to this early-twentieth-century form of spam, was irritated at first: a letter from an unknown Indian containing crazy-looking theorems and no proofs at all. But as he went about his day, Hardy couldn't quite forget about the script:

At the back of his mind [...] the Indian manuscript nagged away. Wild theorems. Theorems such as he had never seen before, nor imagined. A fraud of genius? A question was forming itself in his mind. As it was Hardy's mind, the question was forming itself with epigrammatic clarity: is a fraud of genius more probable than an unknown mathematician of genius? Clearly the answer was no. Back in his rooms in Trinity, he had another look at the script. He sent word to Littlewood that they must have a discussion after hall...

Apparently it did not take them long. Before midnight they knew, and knew for certain. The writer of these manuscripts was a man of genius.

From the foreword by C. P. Snow to Hardy's A Mathematician's Apology

Hardy invited Ramanujan to Cambridge, and on March 17, 1914 Ramanujan set sail for England to start one of the most fascinating collaborations in the history of maths. Right from the start the pair produced important results and Ramanujan made up for the gaps in his formal maths education by taking a degree in Cambridge. Perhaps the most famous story to emerge from this period has Hardy visiting Ramanujan as he lay ill in bed. Hardy complained that the number of the taxi he had arrived in, 1729, was a boring number, and that he worried this was a bad omen. "No," Ramanujan replied, apparently without hesitation. "It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways":

  \[ 1729 = 1^3 + 12^3 = 9^3 + 10^3. \]    

Unfortunately, Ramanujan's sickness wasn't a one-off. His health had always been feeble, and the cold weather and unaccustomed English food didn't help. Ramanujan decided to return to India in 1919 and died the following year, aged only 33. He is still celebrated as one of India's greatest mathematicians.

You can find out more about Ramanujan's mathematics in these Plus articles:

We have also recently reviewed the famous essay A mathematician's apology by Ramanujan's collaborator G.H. Hardy.

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:


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 $\sqrt2$ 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.


The idea is beautifully explored by the Argentine author Jorge Luis 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 never to 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

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


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

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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