** Fractals**

Fractals are dimensionally discordant. ~ Benoît Mandelbrot

A fractal is a set of scale-invariant, self-similar, iterative patterns. Fractals are abundant in Nature and can be produced via formulas of complex numbers.

Looking down on a sea shoreline from 100 meters displays a series of arcs where the water has receded from the sand. At a closer scale, such as 10 meters, one’s field of vision is smaller, but the pattern of arcs is self-similar. At 1 meter and 100 cm selfsame iterative patterns recur. A shoreline is fractal.

Fluid turbulence forms fractal patterns – similar, though not quite identical – at numerous scales of observation.

The patterns that tree leaves and branches form are fractal. Romanesco broccoli (pictured) is floridly fractal.

Fractals got their mathematical start in the 17th century, with Leibniz pondering recursive self-similarity, though his thinking was limited to straight lines. 2 centuries later, German mathematician Karl Weierstrass presented the first function and graph that had fractal properties.

Numerous mathematicians became fascinated with the multidimensional properties of fractals and elaborated mathematical constructs which produced them in a wide variety of ways.

Polish-born French American mathematician Benoît Mandelbrot popularized fractals. He was first to use computers to generate fractal images. His Mandelbrot set (pictured) is well-known. It is viewed on a computer as an infinite series of planes with ceaseless sets of self-repeating patterns which vary by location within the set and at different depths (by zooming in). Exploring the Mandelbrot set is like looking through a microscope at an endless universe of patterns that seem simultaneously both natural and otherworldly. (The covers of Spokes books are Mandelbrot-set fractals.)