The Lorenz Attractor: Chaos in Three Equations
In 1963, Edward Lorenz discovered that a simple set of three differential equations — modeling atmospheric convection — produced a trajectory that never repeats. The resulting butterfly-shaped attractor became the icon of chaos theory. Small changes in initial conditions produce wildly different trajectories, yet all of them trace the same strange shape.
See the collection →Fractal Flames: Scott Draves and the Art of IFS
Fractal flames are a class of iterated function systems invented by Scott Draves in 1992. Unlike classical IFS, flames use nonlinear variation functions — sinusoidal, spherical, horseshoe, julia — combined with log-density rendering and gamma correction. The result is an organic, luminous aesthetic that looks like actual fire or nebulae.
See the collection →Gray-Scott Reaction Diffusion: Why Leopards Have Spots
Alan Turing predicted in 1952 that chemical reactions could spontaneously produce spatial patterns from uniform initial conditions. The Gray-Scott model is one implementation of this idea — two virtual chemicals, U and V, diffuse and react at different rates. Change the feed and kill rates and you get spots, stripes, coral, fingerprints, and labyrinthine mazes.
See the collection →Flow Fields: Following the Invisible Wind
A flow field assigns a direction to every point in space. Particles placed in that field follow the directions like leaves on a stream. When the underlying field comes from noise functions, cellular automata, or point vortices, particles trace paths of unexpected beauty — from turbulent eddies to smooth laminar currents.
See the collection →Julia Sets: The Boundary Between Order and Chaos
Pick a complex number c. Iterate z → z² + c starting from every point in the complex plane. Points that escape to infinity are colored by escape speed; points that stay bounded form the Julia set. The boundary between these regions is fractal — infinitely detailed at every scale. Changing c by even a tiny amount transforms the entire shape.
See the collection →Chladni Figures: The Geometry of Sound
Ernst Chladni discovered in 1787 that sand sprinkled on a vibrating metal plate arranges itself into precise geometric patterns. The sand migrates to the nodal lines — places where the plate doesn't move. These patterns are solutions to the 2D wave equation, and their shapes depend entirely on the plate's vibrational frequency.
See the collection →De Jong Attractors: Four Numbers, Infinite Patterns
The Peter de Jong map is defined by just four parameters: x_{n+1} = sin(a·y_n) - cos(b·x_n), y_{n+1} = sin(c·x_n) - cos(d·y_n). Iterate millions of points and accumulate their positions in a histogram. Different values of a, b, c, d produce wildly different shapes — from hairy knots to elegant spirals.
See the collection →Möbius Transformations: Circles Into Circles
A Möbius transformation maps the complex plane via f(z) = (az+b)/(cz+d). These are the conformal maps that preserve angles and map circles to circles (or lines). When you apply them to a grid, straight lines become arcs; arcs become arcs with different curvatures. The result is the architecture of hyperbolic geometry made visible.
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