Illustrated Physics
physics, drawn
Concept

Fractal dimension — where it appears

The power with which an object's mass or number of parts grows with its size, which for a fractal is not a whole number. A cluster grown by diffusing particles in a plane has dimension 1.71, so it grows emptier as it grows larger.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The set an orbit that never repeats settles onto. 24,000 successive positions of one orbit of the map x' = 1 − 1.4x² + y, y' = 0.3x, after five hundred steps of transient have been discarded. Nearby points separate at e^0.4188 per step, so the orbit is unpredictable in the way a doubling error measures; and every one of the 24,000 points lies inside a box 2.558 by 0.767, a diagonal of 2.670, so it is going nowhere. Those two statements are not compatible with a smooth stretching: something has to bring the separated points back, and the bringing back is the visible fold at the left-hand end. The curve is not a curve. Every strand of it is a bundle of strands at any magnification, which is what an area contraction of 0.3 per step leaves behind when the stretching along the other direction is e^0.419. The 24,000 points paint 7,352 distinct marks at the resolution this is drawn at, which is itself a measurement of how little of the plane the set occupies.

The fold that has to be there

Two trajectories that separate exponentially, in a region they can never leave, are being asked to do two incompatible things. The resolution is that the motion is folded back on itself over and over, and the object that survives infinitely many foldings is neither a curve nor a patch of surface — it has a dimension between the two, and the number can be measured two entirely different ways.

mechanics · Chaos A cluster grown by random walkers. 3000 particles, each released far away to wander at random on a square grid until it first touches the cluster and sticks there, starting from one seed at the centre. Colour marks the order of arrival: the first third (violet), the second (blue), the last (red). The cluster is 220 grid steps across, branched and open, and almost every late arrival is at the outer tips: walkers wander into the outer branches long before they can find their way into the gaps between them. The gaps never fill.

The cluster that grows where the walkers arrive

Let particles wander in one at a time from far away, each sticking where it first touches a growing clump. The rule has nothing in it about shape, and the clump that results is a lacy, branched thing whose mass grows more slowly than its area, so it gets emptier the bigger it gets. The reason is that a random walker reaches the outermost tips long before it can find its way into a gap between branches. The same arithmetic — the probability that a walker arrives is a solution of Laplace's equation — shapes electrodeposits, lightning-like discharges in plastic, fingers of water pushed into oil, and colonies of bacteria short of food.

thermodynamics · Diffusion

Named alongside it

The objects these essays reach for when they reach for this one.

AttractorBox countingChaosDeterminismDiffusionDissipationHarmonic measureInstabilityKaplan yorkeLaplace equationLyapunov exponentPhase space volume

All concepts