math & fractals
The Great Wave off Kanagawa by Katsushika Hokusai captures the imagination with its daring and daunting features. The color woodcut is a master piece in art but is there more to the picture than mere esthetic? Is there mathematical secrets locked inside the photo that would not be discovered until a hundred years later? The picture is blue waves crashing in front of the sacred Mount Fuji, tossing and turning boats on the water. The tips of the big blue waves have scaled smaller waves making the whole. It would not be until Benoit Mandelbrot entered the math scene that fractal geometry would become a part of mainstream mathematics. When Mandelbrot looked at the work of Hokusai he noticed the recurring geometric pattern congruent with fractal geometry.
Fractals are repeating geometric patterns that are infinite in the strictest sense. If we can image a fractal as a pattern on a shirt, than the shirt take an infinite regression to great detail. No matter how close or far away (if one could zoom into the design) from a fractal the pattern is the same. For example with the great wave, the smaller white waves are devised from the large wave, scaled and repeated. The wave was described by the Mandelbrot Set Zn+1=Zn2+C. Where C is in the complex plane, C=-1 and bounded inside infinity. As one can see from the painting, it does not follow strictly the mathematical definition as the waves stop recurring after two waves.