In every field, the greatest innovation occurs when the field arises. In the case of fractal geometry, the greatest innovation occurred when my lifetime work extended the scope of quantitative science to include roughness.
Roughness is ubiquitous but used to be a ‘quality,’ something that everyone could feel but no one could measure. The innovation is that - at long last - it has become a quantity. For time, pitch and colour, this rare transition from qualitative to quantitative relies on the pendulum or uniform motion of a point round the circle.
For roughness, the proper measure was buried in mathematical esoterica that had been misunderstood as unrelated to anything real or anything previously known (In fact, it had already been anticipated, from time immemorial by art.)
A more specific innovation? In terms of popular appeal - ranging from specialists to the masses - surely the Mandelbrot Set wins.
I become a scientist by a family tradition and because, aged 19, I fell in love with geometry in its oldest, most concrete or sensual form: an elusive point where formula and picture meet on even terms, theory meets the real world, and mathematics and hard science meet art, so that their worth and beauty shine beyond the worlds of experts.
Benoit Mandelbrot is Battelle fellow, Pacific Northwest National Laboratory, Sterling professor emeritus of mathematical sciences, Yale, IBM fellow emeritus (physics) at the IBM Research Center.