In 1963 the meteorologist Edward Lorenz reduced convection — a fluid layer heated from below — to three ordinary differential equations, and found that a fully deterministic system can be forever unpredictable: the butterfly effect. Every trajectory is unique, yet all of them weave the same two-winged figure, the Lorenz attractor.
This study renders the attractor as orbit density: tens of thousands of particles ride the equations in real time on the GPU, and their trails accumulate like a long photographic exposure. Brightness is literally the time the flow spends in each region — the attractor drawing its own portrait.
twin spiral galaxies — the attractor seen from directly aboveσ 10 · ρ 28 · β 8/3 · az 0 · el 84 · palette storm
MotifLorenz attractor / Rayleigh–Bénard convection / orbit density
MethodA small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer that accumulates particle trails as orbit density. The images were selected through parameter exploration — sweeping the model's regimes and choosing each frame by eye.
ObservationBrightness maps the flow's residence time; the two wings — the two senses in which the convection roll can turn — trade the orbit forever, and every jump between them is the roll reversing. Rotating the projection turns the butterfly into a pair of touching spiral galaxies.
ReferenceEdward N. Lorenz, "Deterministic Nonperiodic Flow," Journal of the Atmospheric Sciences, vol.20, 130-141 (1963).
This is not a scientific simulation result, but a visual interpretation of the phenomenon.
PARAMETERS EXPLORED
parammeaningeffect on the image
ρRayleigh number ratio — the strength of the thermal drivethe regime itself: steady ⇄ chaos ⇄ periodic windows. The aperture of the eyes, the height of the wings (ρ 99.65 opens a periodic torus knot)
σPrandtl numberhow tightly the wings coil — σ 16 for an engraved rosette, σ 26 for the paired owl-eyes
βgeometry of the convection cellthe spread of the wings — β 0.8 for goggles, β 5.5 for the lit blades of a pupil
view (az, el)projection directionbutterfly (front) ⇄ blade (edge-on) ⇄ twin galaxies (from above). The viewpoint is a creative axis in its own right
particles × decaydensity of the orbit bundle × exposure timea few bundles read as engraved rings; a great many smooth into a nebula
Each image below records its exact parameter set.
SELECTED STILLS — 5
twin spiral galaxies — from directly aboveσ 10 · ρ 28 · β 8/3 · az 0 · el 84 · palette storm
engraved lace — a pose-jump climaxρ 28→64 · el 74 · engraved trails · palette storm
torus knot — a periodic windowρ 99.65 · periodic orbit · az 0 · el 0
weaver variant — the two roll senses in house coloursσ 10 · ρ 28 · β 8/3 · palette weaver
The two tints follow the sign of x in the model — physically, the two directions the convection roll can rotate. Each jump between wings is the roll reversing, so the colour boundary you see is a real dynamical divide, not a decorative one.
Brightness follows residence time, the attractor's own natural measure: the glowing "eyes" are where orbits spiral slowly around the unstable steady states, and the bright seam down the middle is where both wings fold together.
The colouring is an interpretive mapping based on the physical meaning of the model's variables — an artistic approximation, not a measurement.
the two roll senses, in house colours — steel and amberσ 10 · ρ 28 · β 8/3 · palette weaver
Palette wing — hue = sign of x (roll direction) · luminance = residence time · the eyes and the folding seam glow brightest.
REFERENCES
Edward N. Lorenz, "Deterministic Nonperiodic Flow," Journal of the Atmospheric Sciences, vol.20, 130-141 (1963).
