STUDY #05 · 2026 · IN OBSERVATION

Lorenz Attractor

A model-driven visual study of deterministic chaos.

MOVING IMAGE — ORBIT DENSITY σ 10 · ρ 28 · β 8/3 · view spin

WHAT IS THIS

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.

engraved lace — the orbit bundle at a pose-jump climax ρ 28→64 · el 74 · engraved trails · palette storm

Motif Lorenz attractor / Rayleigh–Bénard convection / orbit density

Method A 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.

Observation Brightness 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.

Reference Edward N. Lorenz, "Deterministic Nonperiodic Flow," Journal of the Atmospheric Sciences, vol.20, 130-141 (1963).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

A NEW CHAPTER

From fields that fill the frame to a single orbit that weaves it.

Studies #01–04 — fields Study #05 — orbit

Subject a field everywhere at once — chemistry, sound, heat one trajectory in time, tracing a shape it never leaves

Brightness concentration or amplitude of the field residence time — where the orbit lingers, a long exposure

The hidden force a reaction, a resonance, a temperature determinism itself — predictable in law, unpredictable in fact

PARAMETERS EXPLORED

param meaning effect on the image

ρ Rayleigh number ratio — the strength of the thermal drive the regime itself: steady ⇄ chaos ⇄ periodic windows. The aperture of the eyes, the height of the wings (ρ 99.65 opens a periodic torus knot)

σ Prandtl number how tightly the wings coil — σ 16 for an engraved rosette, σ 26 for the paired owl-eyes

β geometry of the convection cell the spread of the wings — β 0.8 for goggles, β 5.5 for the lit blades of a pupil

view (az, el) projection direction butterfly (front) ⇄ blade (edge-on) ⇄ twin galaxies (from above). The viewpoint is a creative axis in its own right

particles × decay density of the orbit bundle × exposure time a 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 above σ 10 · ρ 28 · β 8/3 · az 0 · el 84 · palette storm

engraved lace — a pose-jump climax ρ 28→64 · el 74 · engraved trails · palette storm

torus knot — a periodic window ρ 99.65 · periodic orbit · az 0 · el 0

weaver variant — the two roll senses in house colours σ 10 · ρ 28 · β 8/3 · palette weaver

coexisting blades — a lit pupil ρ 45 · β 5.5 · edge-on · palette storm

SIGNATURE — THE PUPIL BREATHES

One always-bright line, and two eyes that open and close.

As ρ rises and falls the dark centres dilate and close while the whole figure turns; the burning fold where both wings meet is the one line that never dims.

In the hero above this is the loop's slow breath; in the Interactive Study below it is the Drive knob — turn it and watch the pupils open.

twin spiral galaxies — the two eyes and the bright fold-seam σ 10 · ρ 28 · β 8/3 · az 0 · el 84 · palette storm

COLOUR = ROLL DIRECTION

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 amber σ 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).

INTERACTIVE STUDY

A small live model runs the same idea in your browser — the Lorenz system, integrated in real time, its orbits accumulating into density just like the works above. It is a deliberately simplified instrument, reduced in resolution and particle count, separate from the full GPU engine used to author the finished works. Turn the drive up and watch steady convection tip into chaos; lift the view from the butterfly to the twin galaxies.

SIMPLIFIED INSTRUMENTLORENZ ORBIT DENSITY · LIVE

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.