Watch crystals grow in real time using Diffusion-Limited Aggregation (DLA) — a fractal growth process that models snowflakes, lightning bolts, mineral deposits, and coral reefs.
Particles perform random walks in 3D space. When a wandering particle touches the growing crystal it sticks permanently. The fractal branching structure that emerges mirrors real-world dendritic crystal growth.
Press play to start growing. Adjust particle speed and emission rate. Click to add a seed point and watch new branches sprout from that location.
The same DLA mathematics describing crystal growth also explains the shapes of lightning bolts, river deltas, coral reefs, and even neuron growth patterns in the brain.
This simulation grows a three-dimensional fractal crystal using diffusion-limited aggregation (DLA). Particles released far from the cluster perform a lattice random walk, and whenever one lands beside the existing aggregate it sticks with a probability set by the stickiness control. Because branch tips intercept wandering particles before they can reach the interior, the cluster develops the screened, dendritic, self-similar shape characteristic of snowflakes, mineral deposits and electrodeposits.
A seed sits at the origin and walkers spawn on a sphere of the chosen walk radius. Each frame they hop to a random neighbouring lattice site; on touching the aggregate they attach with probability equal to the stickiness value. The lattice (cubic 6-neighbour, hexagonal, or FCC 12-neighbour) sets the bonding geometry, and each new particle is coloured by its layer depth from the seed, from deep violet through cyan to a near-white tip.
Four sliders drive the model: Stickiness (0.1-1) is the attachment probability, Walk radius (5-40) sets the spawn sphere, Batch size (1-30) is how many walkers stay active, and Particle size (0.2-1.2) scales the spheres. The Cubic, Hexagonal and FCC buttons switch lattice type, Restart reseeds, and you drag to rotate and scroll to zoom the auto-rotating view.
Classic 2D DLA clusters have a fractal dimension of about 1.71, meaning they fill space far less densely than a solid disc. The model, introduced by Witten and Sander in 1981, describes growth wherever transport is limited by diffusion, including lightning, river deltas and bacterial colonies.
DLA is a growth process in which particles wander randomly and irreversibly stick on contact with a growing cluster. Because new arrivals are far more likely to hit the exposed tips than to diffuse deep into the structure, the cluster grows into a branching, porous, fractal shape rather than a compact solid.
A seed is placed at the origin and walkers are spawned on a sphere of the walk radius. Each step they move to a random neighbouring lattice site; when a walker is adjacent to the aggregate it attaches with the stickiness probability, otherwise it keeps walking and is discarded if it strays too far. The process repeats up to a cap of around 2000 particles.
Stickiness is the probability that a walker bonds when it touches the cluster: lower values let particles slide around tips before sticking, producing denser, more rounded forms, while higher values lock branches in early for spindlier growth. The lattice buttons change which neighbours count as contacts, cubic giving six directions, hexagonal ten and FCC twelve, altering the crystal symmetry.
It is a qualitatively faithful caricature rather than a quantitative one. Real crystals also involve surface energy, temperature gradients and detachment, none of which are modelled here. DLA nonetheless captures the essential physics of transport-limited dendritic growth, which is why its branching patterns resemble genuine mineral and electrochemical deposits.
The cause is geometric screening. Protruding tips reach out into the surrounding space and capture diffusing particles before those particles can travel into the sheltered interior valleys. This positive feedback amplifies any small bump into a branch, leaving the inner regions starved and producing the open, self-similar fractal seen on screen.
This simulation models crystal growth using Diffusion-Limited Aggregation (DLA), a mathematical process in which particles performing random walks in three-dimensional space irreversibly attach to a growing cluster on contact. The result is a self-similar fractal structure whose branching geometry emerges purely from the statistics of diffusion, reproducing the open dendritic forms seen in snowflakes, mineral deposits, and electrochemical plating. Users can observe how stickiness probability, lattice geometry, and spawn radius all reshape the final crystal architecture.
DLA was introduced by Witten and Sander in 1981 and has since become a foundational model in condensed-matter physics, materials science, and geomorphology. Real-world systems governed by the same mathematics include frost patterns on glass, the branching of river deltas, coral reef growth, and the formation of manganese oxide dendrites in sedimentary rock.
Diffusion-Limited Aggregation is a stochastic growth process in which particles execute random walks and permanently bond to a cluster the moment they touch it. Because transport to the cluster is limited by diffusion rather than by reaction rate, the mathematical term "diffusion-limited" distinguishes it from reaction-limited growth where particles bounce off many times before bonding. The slow interior of the cluster is starved of arriving particles, so growth concentrates at the exposed tips and produces a fractal branching pattern.
The Stickiness slider (0.1 to 1.0) sets the probability that a walker bonds when it first touches the aggregate: low values allow particles to slide past tips and fill gaps, yielding denser clusters, while values near 1.0 lock every contact immediately and produce the spindliest fractals. Walk Radius determines the size of the sphere from which walkers are launched; a larger radius gives more time for diffusion and accentuates tip-screening. Batch Size controls how many walkers are active simultaneously, trading accuracy for speed. The Cubic, Hexagonal, and FCC buttons switch the bonding lattice, and Restart reseeds the simulation from a single particle at the origin.
The branching is caused by geometric screening: any protrusion that extends slightly ahead of its neighbours intercepts diffusing particles before they can penetrate the sheltered interior gaps. This is a positive-feedback instability -- the further a tip reaches, the greater its capture cross-section, so it grows faster still while the hollows behind it stagnate. The same instability appears in the Mullins-Sekerka theory of solidification fronts, confirming that branching is a universal consequence of diffusion-limited transport rather than a detail of any particular material.
In two dimensions the DLA cluster has a fractal (Hausdorff) dimension of approximately 1.71, meaning it fills space far less efficiently than a compact disc (dimension 2). In three dimensions the value rises to roughly 2.5, still well below the embedding dimension of 3. These numbers are measured by counting how the number of occupied sites N scales with the radius R of the cluster: N ~ R^D where D is the fractal dimension. The non-integer value reflects the self-similar, scale-free branching architecture and is one of the earliest examples of a fractal dimension measured in a physical growth experiment.
Snowflakes grow by water-vapour molecules diffusing through air and freezing onto the ice surface, a process transport-limited in exactly the DLA sense. The six-fold symmetry of ice crystals arises from the hexagonal lattice of the water molecule, while the branching arms reflect the Mullins-Sekerka instability of the freezing front. Switching this simulation to the Hexagonal lattice mode produces clusters with visible six-fold symmetry, demonstrating how lattice geometry dictates the macroscopic crystal habit. Real snowflakes are further shaped by temperature and humidity gradients the flake encounters as it falls, which is why no two are identical.
DLA captures the transport-limited regime faithfully but omits several phenomena present in real mineral growth: surface energy anisotropy, thermal fluctuations, detachment and dissolution, solute depletion fields, and convection. Quantitatively accurate crystal-growth models such as phase-field methods or kinetic Monte Carlo simulations include these effects and can reproduce specific crystal habits. DLA remains a qualitatively correct and computationally cheap archetype that explains why deposit morphologies ranging from electrochemically grown zinc fractals to manganese dendrites in agate share the same open, branching form.
Thomas Witten and Leonard Sander published the DLA model in 1981 in a Physical Review Letters paper titled "Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon." The paper was motivated by the need to explain the branching morphology of electrochemical zinc deposits and aerosol agglomerates, which classical compact-growth models could not reproduce. Witten and Sander's computer simulations showed that random-walk attachment alone generates the observed fractal structures, making DLA one of the first computational physics models to reveal a universal scaling law in non-equilibrium growth.
The list of DLA-like systems is remarkably broad. Dielectric breakdown (lightning and spark discharge) follows a Laplacian growth equation identical in form to diffusion, producing the same branching patterns. River-delta networks, Hele-Shaw viscous fingering, coral-reef skeletal growth, bacterial colony morphology under nutrient-diffusion limitation, and neurite outgrowth in neurons all display fractal dimensions close to the DLA value. The unifying theme is that whenever a scalar field (concentration, electric potential, pressure) diffuses to a growing interface, the interface is unstable to tip-splitting and branching.
Engineers exploit and control DLA-type growth in several fields. In electrochemistry, understanding DLA instability guides the design of battery electrodes: lithium dendrites that form on anodes during fast charging follow DLA kinetics and can short-circuit cells, so electrolyte additives and structured electrode surfaces are engineered to suppress branching. In materials science, porous DLA-like aerogels are deliberately grown for their high surface-area-to-volume ratio, useful in catalysis and thermal insulation. In antenna design, fractal geometries inspired by DLA offer wideband performance in compact form factors. Microfluidic devices use controlled viscous fingering (a DLA analog) to mix fluids that would otherwise remain laminar.
Current research extends DLA in several directions. Chemists study off-lattice DLA with realistic inter-atomic potentials to bridge the gap between the toy model and quantitative crystal-growth prediction. Physicists investigate multi-particle DLA variants and the crossover between DLA and Eden (reaction-limited) growth as a function of stickiness, exactly the parameter this simulation exposes. In battery science, machine-learning models trained on DLA simulations are used to predict dendrite nucleation sites before they appear experimentally. In biophysics, DLA frameworks model biofilm architecture and the branching of blood-vessel networks during angiogenesis, opening pathways to anti-cancer strategies that starve tumours by disrupting their diffusion-driven vascular growth.