Amethyst is a violet variety of quartz (SiO₂) that forms as silica-bearing fluid slowly deposits material onto a growing crystal. This companion swaps the 3D version's finished, hand-sculpted crystal for the actual growth process behind it: a diffusion-limited aggregation (DLA) model, the standard toy physics for how a crystal accretes particle by particle from a supersaturated solution.
Each free "ion" performs a random walk — a discrete approximation of Brownian motion, whose step size follows the Einstein relation (diffusion coefficient D ∝ temperature). When a walker drifts within bonding range of the existing cluster it sticks with a probability that combines a base kinetic sticking factor with a six-fold angular bias, modelling the anisotropic attachment rates along quartz's real hexagonal crystal axes:
step = step0 · sqrt(T / T0) (Brownian step ~ sqrt(D), D ∝ T)
axial = 0.5 + 0.5 · cos(6θ) (six-fold hexagonal bias, θ = angle from cluster centre)
P_stick = clamp( S_base · (1 − A·(1 − axial)), 0.02, 1 )
- Temperature — sets the random-walk step size (kinetic mobility of the ions).
- Supersaturation — sets how many ions are diffusing through the solution at once, which sets how fast the cluster accretes.
- Sticking probability — the base kinetic barrier to attachment; lower values grow denser, more compact clusters, higher values grow sparser, more dendritic ones.
- Hexagonal anisotropy — how strongly attachment favours the six crystallographic growth axes; at 0 the cluster grows as an isotropic blob, near 1 it grows visibly faceted spikes.
- Fractal dimension — estimated live from cluster mass vs. radius (D = ln N / ln R); classic off-lattice DLA converges to D ≈ 1.71 in 2D, and this readout drifts toward it as the cluster grows large.