Technique · Fields
Julia and Mandelbrot lace
The escape-time fractal z squared plus c, bevelled and lit like carved enamel, with orbit-trap colour in the filigree.
Julia and Mandelbrot lace is a field drawn for every pixel by the graphics chip. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- Orbit traps colour a pixel by the closest approach of its orbit to a chosen point, curve or shape; the idea grew from Clifford Pickover's fractal stalks of the 1980s and gives the ornament inside the lace.
- Peitgen and Richter's book The Beauty of Fractals (1986), written alongside the travelling exhibition Frontiers of Chaos, carried pictures like these to a wide public.
- The Julia set of z squared plus c is connected exactly when c belongs to the Mandelbrot set; otherwise it falls apart into dust.
- Gaston Julia published his study of these sets in 1918, aged 25, in a long paper on iterating rational functions; Pierre Fatou reached many of the same results independently.
- The first picture of the Mandelbrot set was drawn in 1978 by Robert Brooks and Peter Matelski; Benoit Mandelbrot studied it by computer at IBM from 1980.
- Mitsuhiro Shishikura proved in 1998 that the boundary of the Mandelbrot set has dimension 2, even though it is a curve.
The mathematics
- Once the orbit passes a radius of 2 (and the size of c) it must escape to infinity, so a larger cut-off like the 500 used here only matters for smoothness: the smooth count n - log2(log|z| / log 500) removes the bands between whole steps, because |z| roughly squares at every step.
- Orbit points are complex numbers, and squaring one doubles its angle and squares its length: that doubling of angle is what folds the plane into the repeating spirals of the picture.
- This piece picks c on the boundary of the Mandelbrot set's main cardioid, 0.5 e^(i a) - 0.25 e^(2 i a), at angle a = 2.10, pulled inward by 0.020; just inside the set the Julia set stays connected and ornate.
- The bevel lights the smooth escape count as a height, using its slope across the screen, so the carved look needs no 3D geometry at all.
In art
- Kerry Mitchell's Fractal Art Manifesto of 1999 argued that fractal images are art because the artist chooses the formula, the view and the colours, the way a photographer chooses a subject and a lens.
- Ornament has always loved self-similar curls: the scrolls of Baroque carving and the whorls of Celtic metalwork echo at every scale.
More ways to use it
Related techniques
Nebula and stars · Liquid · Droplets · Network · Terrain and maps · Neuron · Order to chaos · Fractal tree