Phase space — deterministic chaos
The figure.
The dark object on the front page is a real strange attractor, computed live in your browser. This note explains what it is, what the brightness measures, and where the science ends and our rendering begins.
The note
Back to the front page ↗What is shown
Every element on the card declares something.
The plane is not physical space. Each point (x, y) is one complete state of the system at a single instant. The figure is the system’s own history, superimposed on itself. You are looking at time, not matter.
The regime it runs in: the rule is exact, and the long-run outcome is not predictable.
The accumulated orbit — millions of consecutive states, each computed from the one before it.
How to read the brightness. Bright is not decoration: it is where the system spends its time.
The four parameters of the rule, live. They drift slowly, so the figure never repeats.
Where it comes from
An object with a name, not a decorative invention.
It is a strange attractor from the family of iterated maps attributed to Clifford A. Pickover, who popularised them in his work on computation and the visualisation of pattern (Computers, Pattern, Chaos and Beauty, 1990), and catalogued extensively by Paul Bourke. The whole system is two lines:
yn+1 = sin(b·xn) + d·cos(b·yn)
a = −1.4 b = 1.6 c = 1.0 d = 0.7
Take a starting point — here (0.1, 0.1) — put it through the rule, take the result, and put it through again. Millions of times. The values above are the classic Clifford regime.
The science
Four concepts, each by its name.
A discrete-time dynamical system. Time advances in steps, and each step applies the same function. There are no differential equations: it is repeated application of one rule. The sequence of states is called the orbit.
Start from almost any initial point inside the basin of attraction. After a transient, the orbit converges onto one specific set and never leaves it again: it wanders inside it forever. That set is the attractor, and the system is attracted to it. A fixed point and a limit cycle are simple attractors. This one is not.
The attractor is fractal: it has detail at every scale and a non-integer dimension. Zoom in and more layers of filament appear, indefinitely. Ruelle and Takens coined the term in 1971, studying turbulence. “Strange” describes the geometry, not the behaviour.
The rule contains no randomness. The same parameters and the same starting point give exactly the same figure, on any machine, every time. And yet two starting points arbitrarily close together separate exponentially (a positive Lyapunov exponent). The system is fully determined and, in the long run, not predictable. Lorenz found this in 1963, and it is why weather forecasting has a horizon. Determinism does not imply predictability.
The orbit does not visit the attractor evenly: some regions are visited far more often than others. Counting visits per pixel over millions of iterations approximates the natural invariant measure — a real object of ergodic theory. The brightness in the figure is that measure, directly. The bright filaments are where the system spends most of its life. The image is a measurement, not an illustration.
What is science and what is rendering
The boundary, said out loud.
The map, the attractor, the deterministic chaos and the invariant measure. And the computation: the browser genuinely iterates ~44,000 times per frame and accumulates density in a Float32 buffer. There is no pre-recorded image and no video.
brightness = 1 − e−(density × exposure) — a saturating curve, the same form as the response of photographic film or a sensor, so heavily visited regions keep their detail instead of clipping to flat white.
A gamma of 0.58 lifts the faint filaments.
A decay of 0.978 per frame is a forgetting factor, so the accumulator follows the current parameters instead of saturating with the whole history.
The drift of a b c d is a presentation choice. The attractor is a function of its parameters; moving them changes the geometry, sometimes smoothly and sometimes through a bifurcation. Drifting them slowly walks a continuous family of attractors. Every frame is a real attractor; the walk is ours. That is why it never loops.
The Clifford map does not model any physical system. It is a constructed map, built to produce rich attractors. Lorenz’s came from convection equations; Hénon’s from a simplified Poincaré section. Clifford’s came from exploring which combinations of sine and cosine produce structure. It is the canonical example of deterministic chaos, not a model of a specific physical system.
Where this mathematics does appear
Strange attractors do govern real systems.
Atmospheric convection. The reason weather prediction has a limit.
Population dynamics.
Chaos in real, physical electronics.
The route to turbulence.
Arrhythmia; and EEG in neuroscience.
The laboratory case.
Why it is on this page
The figure is the argument, not an ornament beside it.
The front page claims software that does not break. That is a claim about determinism: that behaviour is engineered, not hoped for. The figure shows three things, and those three things are the claim:
What looks unpredictable is governed exactly. There is no randomness at any point.
The order is not imposed from outside: it emerges from the rule. Nobody drew that shape; the law produces it.
It is bounded. The orbit is chaotic but never escapes. An engineering standard does not remove complexity: it bounds it.
And, for precision: the connection is rhetorical, not a proof. The figure does not prove that our software does not break. It is a deliberate choice of visual argument, and a fair one, because the object on screen genuinely has the property the sentence claims. The difference from decoration is that here the object is the argument.
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