Time and again, pure mathematics displays an astonishing quality. A piece of mathematics is developed (or discovered) by a mathematician who is, often, following his or her curiosity without a plan for meeting some identified need or application. Then, later, perhaps decades or centuries later, this mathematics fits perfectly into some need or application.

I have recently been interested in this idea, particularly as research funding bodies have been increasingly asking researchers to predict the impact of their research before it is funded, and research quality is being measured partly by its short term impact. On this issue I spoke to more and more research mathematicians and a pattern started to emerge: mathematicians tend to think of this aspect of mathematics as axiomatic, and generally come up with one of the same three examples to justify this position – number theory in cryptography, logic in computing and complex numbers in fluid mechanics.

This made me a little concerned. If every research mathematician you speak to comes up with the same few examples, is the story going to be sufficiently convincing?

I decided there must be mathematicians out there who have less well known examples of mathematics having impact that could not have been planned by the original discoverer – unplanned impact of mathematics. This might be mathematics pursued for curiosities sake, or that had a direct application and was later found to have a different, unexpected one.

Acting in my position as a member of the Council of the British Society for the History of Mathematics (BSHM), I put out a call for further examples. The BSHM has charitable aims around promoting awareness, knowledge and study of the history of mathematics. I spotted a win-win: BSHM could raise awareness of the history of our subject and, in turn, provide researchers with a greater range of examples to call on.

This began a slightly unreal process, particularly when one of my contributors, Edmund Harriss, was able to put me in touch with Nature, who were interested in taking a set of seven contributions.

The field of topology provides an illustrative example. This was started by Euler and studied for 250 years as a purely theoretical discipline before, in the last two decades, finding applications as diverse – and alien to Euler – as DNA, galaxy formation and robotics. These applications rely on the 250 years of pure research, but those advances would not have been made if the researchers had to justify the planned impact before studying their mathematics.

In technology, quaternions, a 19th century discovery which seemed to have no practical value, have turned out to be invaluable to the 21st century computer games industry, while work on the best way to stack oranges started by Kepler in 1611 is essential to modern telecommunications.

Einstein’s theory of relativity, which seemed to come as a spark of genius from nowhere, nevertheless drew on abstract geometry developed half a century earlier. Fourier’s theory of vibrating strings, via very abstract mathematics in the 20th century, has now yielded new insights into quantum physics.

Gambling on 16th century dice games led to a discovery in mathematical probability that is crucial to the insurance industry, while a recent insight into a quantum theory thought experiment has unexpectedly found applications in the outbreak of viral disease and the risks associated with stock market volatility.

These seven examples, contributed by Julia Collins (topology), Mark McCartney and Tony Mann (quarternions), Edmund Harriss (stacking oranges), Graham Hoare (Einstein’s geometry), Chris Linton (Fourier analysis), Juan Parrondo and Noel-Ann Bradshaw (Parrondo’s paradox) and me (dice games), appear in the 14th July 2011 issue of Nature. Go and get a copy and see for yourself!

I’d still like to hear more examples of mathematics displaying this astonishing quality, and I am exploring options to distribute these. You can send me yours – the details are on the BSHM website.

Update 22:15: Link to the article in Nature. Hear me talking about this on the Nature Podcast – you can access the podcast using the following player. The unplanned impact bit starts at 08:44 (though listen to the rest, of course!).

Update 15/07/11 08:20: I recorded a segment for podcast the Pod Delusion, 15th July 2011. Listen to it all, of course, but my bit starts at 19:10.

Update 18/07/11 08:36: Edmund Harriss joined Samuel Hansen and I and we spoke about The unplanned impact of mathematics for Math/Maths Podcast 56: The unplanned impact of mathematics.

Update 25/01/12: Tim Harford wrote a column in the FT on this on 24th September: New ways with old numbers (also on FT.com).