Fractals
Complex-plane escape-time systems
What it is
Fractals in this category come from iterating a function on complex numbers, pixel by pixel, and recording whether (and how fast) each point escapes to infinity. The image is either the set of points that never escape (the black "prisoner set") or a coloring of everyone else by how quickly they got away. Small changes to the iteration formula - an added exponent, an absolute value, a memory term referencing the previous step - produce entirely different-looking families from the same basic technique.
History
Gaston Julia and Pierre Fatou studied iterated complex functions in the 1910s-20s, decades before computers could visualize what they were describing. Benoit Mandelbrot, working at IBM in the late 1970s and 80s, used early computer graphics to render the set that now bears his name, coining the term "fractal" (from Latin fractus, "broken") in 1975 to describe self-similar, infinitely detailed shapes. The resulting image became one of the most recognizable outputs of computational mathematics and helped popularize chaos theory far beyond academia.
Main Contributors
Gaston Julia, Pierre Fatou, Benoit Mandelbrot, and later explorers such as Michael Barnsley (fractal compression and iterated function systems) and John Milnor (complex dynamical systems theory).
Latest Trends
Real-time GPU shader rendering now makes smooth infinite zooming interactive rather than a multi-hour batch render. Arbitrary-precision "deep zoom" renderers use perturbation theory to explore fractals far beyond what ordinary floating-point math allows. And generative-art communities increasingly treat parameter families - Multibrot, Burning Ship, Phoenix - as a design space to explore, discovering new visual territory from tiny formula tweaks rather than a single fixed picture.
Mandelbrot / Multibrot
The classic escape-time set, generalized from z²+c to zn+c - a single exponent slider turns one fractal into a whole family.
Phoenix Fractal
Adds a memory term referencing the previous iteration, producing tendrils, feathers, and flame-like sprays instead of smooth bulbs.
Burning Ship
One absolute value on the imaginary term turns Mandelbrot's bulbs into jagged flame and mountain-like silhouettes. Built for deep zooming.