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The Fourier Transform is used in a wide range of disciplines and activities - it can even save your life! Find out how in Plus Magazine's two articles on the mathematics of tomography (the second article discusses the mathematics behind tomography).

For the musically inclined you can find out how the Fourier transform can lead to a career in audio software engineering

Underground Mathematics has some activities for school students interested in finding out more about the Fourier Transform.

For a short history of the Fourier Transform see this Guardian article.

If you're confused about what the Fourier Transform actually is - Stack Overflow has the answer.

Terence Tao has also written a good introductory paper on the Fourier Transform.

The Fourier Transform can be applied to many things:

http://xkcd.com/26/ Fourier transform of cat

One of the reasons it's so powerful is that it breaks down any periodic function into a combination of sines and cosines. You can also do this for orbits with Ptolemy's system of epicycles and deferents:

 

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