. The above represents a selective but fundamental body of work that modern living mathemati- cians can draw on to produce transforms optimised to perform specific tasks. In 1965, 110 years after the death of Carl Friedrich Gauss, JW Cooley at the IBM laboratories working with John W Tukey of Princetown University published a paper that showed how the Fourier series could be reduced recursively by decimating in time or frequency, an approach similar to Gauss whose work at the time was unknown to them. This became the basis for the fast Fourier transform and is often now cited as the genesis of modern digital signal processing. The Cooley-Tukey algorithm inspired a whole generation of mathematicians to produce fast and faster implementations of the Fourier transform that trade accuracy against the amount of multiplication needed against the number of additions needed against the amount of mem- ory space needed against the number of clock cycles needed against the processing delay and power limitations of the host device. Examples include the Guo and Burrus wavelet based FFT in 1996, the Edelman fast multipole method (based on the compressibility of the Fourier matrix rather than the compressibility of the data), and other options like the Good Physical Layer Connectivity Thomas algorithm, the Rader Brenner algorithm, the Bruun algorithm and the Winograd algorithm. Some of the techniques (for example Guo and Burrus) draw on work done by Benoit Mandelbrot. Mandelbrot's work on fractal geometry has been translated into Basque, Brazilian, Bulgarian, Chinese, Czech, Italian, Portuguese, Rumanian and Spanish SO is unlikely to share the fate of Heinrich Gauss's work on the FFT. Fractals have particularly compelling advantages in image processing, particularly where natural objects (trees and leaves) need to be encoded. We revisit this topic when we look at fractal compression schemes as an extension of existing DCT-based image compression (Mandelbrot maths in mobile phones) in Chapter 9. Benoit Mandelbrot's published wor