Laplacian Growth

Growth that rewards exposure. A harmonic field concentrates attachment chance at tips; each advance screens the recesses behind it. Dendrites and needles are not drawn—they are the irreversible record of boundary-by-boundary competition under Laplace’s equation.

The central field (\phi(\mathbf x)) satisfies Laplace’s equation in the region outside the growing cluster:

Compact form: ∇^2\phi=0.

A function satisfying this equation is called harmonic. Harmonic fields appear whenever a transported quantity reaches a quasistatic steady state without local sources in the domain. Depending on the application, (\phi) can represent concentration, pressure, electric potential or temperature. The mathematical equation is shared; the physical interpretation is not.

A typical growth problem fixes (\phi=0) on the aggregate and a higher value on a distant boundary. The normal field or flux at the interface is proportional to

Compact form: |∇\phi|.

The stochastic growth rule then assigns boundary element (i) a probability such as

Compact form: P_i=\frac|∇\phi_i|^\eta Σ_k\in∂ C|∇\phi_k|^\eta,

where (\partial C) is the growth perimeter and (\eta) controls selectivity.

At (\eta=1), attachment follows the harmonic measure directly. In the dielectric breakdown model, varying (\eta) changes how strongly growth is concentrated at high-field sites. Values above one increasingly favour the most exposed tips. Values below one flatten differences across the perimeter and make growth more compact. The current control help that says “Low = bushy DLA” should be corrected: classical diffusion-limited aggregation corresponds to harmonic-measure growth, closely associated with (\eta=1), while the limit (\eta\to0) approaches more nearly uniform perimeter selection.

Harmonic measure: probability with a global memory

One way to interpret the field is through random walkers. Imagine particles released far from the aggregate and diffusing until they hit it. The probability that a walker first reaches a particular boundary site is the harmonic measure of that site. Protruding tips are easier to hit; deep fjords are shielded by branches in front of them.

This explains why a local attachment event depends on global shape. The selected site does not inspect the entire cluster. The harmonic field has already integrated that geometry into the local gradient. A branch can suppress growth far behind it without direct contact.

This mechanism is screening. Tips compete for flux. A slightly advanced branch gains a larger share, advances again and increases its advantage. Positive feedback produces sparse structures with long active tips and nearly frozen interiors.

Screening is also why the finished aggregate records its history. A branch that lost the competition early may never recover, even if the external conditions later remain unchanged. Growth is irreversible: occupied cells do not ordinarily dissolve or move. The final structure is therefore one realised path through many possible stochastic boundaries.

Diffusion-limited aggregation and dielectric breakdown

Witten and Sander’s 1981 diffusion-limited aggregation model grows a cluster by releasing random walkers that stick on first contact. It generated ramified aggregates with non-integer scaling behaviour and became a foundational computational model of fractal growth.

Niemeyer, Pietronero and Wiesmann’s 1984 dielectric breakdown model replaced explicit walkers with a Laplace field and assigned growth probability proportional to a power (\eta) of the local electric field. This made it possible to tune tip selectivity and relate branching discharge patterns to a common field-driven framework.

AL.Floys is closer to a discrete dielectric-breakdown-style perimeter model than to an explicit walker-based DLA. It repeatedly relaxes a potential on a grid, identifies empty cells adjacent to the aggregate, estimates a local gradient, raises it to eta, and samples growth sites from the resulting weights.

That lineage should guide the public language. The engine can legitimately teach harmonic screening and field-weighted branching. It does not need to call every low-(\eta) preset DLA or every high-(\eta) branch electrical breakdown.

Viscous fingering belongs to the same mathematical neighbourhood, not the same implementation

When a less viscous fluid displaces a more viscous one in a porous medium or Hele–Shaw cell, the interface can become unstable and form fingers. Under suitable approximations, pressure in each fluid obeys Laplace’s equation, and interface velocity depends on its gradient. Saffman and Taylor’s classic experiments and analysis established this instability as a major example of pattern-forming interfacial flow.

The connection is real: both problems involve a harmonic field and a moving boundary. The current Viscous Crown preset, however, is not a fluid simulation. It contains no viscosity ratio, pressure continuity, Darcy law, capillary pressure, interface velocity, volume conservation or physical surface tension. It is a border-seeded stochastic lattice aggregate whose morphology may resemble a fingering front.

The correct editorial formulation is:

Laplacian growth provides a mathematical bridge among several branching phenomena; visual similarity does not make their physical mechanisms interchangeable.

The implemented potential solver

AL.Floys stores a scalar potential on a rectangular grid. Occupied aggregate cells are set to zero. For centre and multi-seed modes, the outer border is held at injectionRate. For border mode, the aggregate begins along part of the bottom edge and the top edge supplies the far-field value.

Empty interior cells are updated by Jacobi relaxation:

Compact form: \phi_i,j^(n+1)=(1)/(4) \left( \phi_i+1,j^(n)+ \phi_i-1,j^(n)+ \phi_i,j+1^(n)+ \phi_i,j-1^(n) \right).

Only one to sixteen sweeps are performed per animation step. The field is therefore approximate and may lag behind a rapidly changing cluster. Potential iterations is not merely a visual quality setting; it determines how closely the sampled weights track the current boundary-value problem.

On reset the engine performs a larger initial relaxation. During growth, several new cells may be attached after one field solve. The potential is not recomputed after each attachment within that frame. At high Growth per step, later events therefore use a field calculated for an earlier boundary. This batching accelerates the simulation but makes the process less faithful to sequential quasistatic Laplacian growth.

How the local growth weight is constructed

A candidate is an unoccupied, non-obstacle cell with at least one occupied cardinal neighbour. The code estimates a gradient from differences between the candidate potential and its free neighbours, adds the candidate’s own positive potential as an extra tip proxy, and raises the resulting magnitude to (\eta).

Several additional factors then modify the weight:

• Sticking probability does not change relative candidate weights; it is a second random gate after a site is selected.
• Surface tension multiplies sites with more occupied neighbours, favouring concavities and compact filling.
• Anisotropy biases the gradient angle toward fourfold or sixfold preferred directions.
• Noise adds multiplicative stochastic jitter.
• a painted scalar bias boosts selected regions;
• an advanced drag tool stores one global direction and boosts gradients aligned with it.

These choices are useful creative controls, but their labels must match their numerical meaning.

“Surface tension” is a compactness heuristic

Physical surface tension penalises interfacial area and acts through curvature-dependent boundary conditions. The implementation does not calculate curvature or capillary pressure. It simply rewards a candidate for having more occupied cardinal neighbours.

That creates a smoothing tendency: filling a notch is favoured over extending a lone tip. The concept is analogous to interfacial regularisation, but the control should be called Compactness bias, Neighbour smoothing or Front smoothing unless a curvature term is implemented.

Anisotropy reveals the grid as part of the model

A square lattice already privileges cardinal directions. The engine adds explicit fourfold or sixfold angular preference. Strong square anisotropy can create competing needles. The nominal hex mode applies a sixfold cosine to a field still evaluated on a square grid with four-neighbour occupancy. It can produce snowflake-like symmetry, but it is not a true hexagonal lattice.

Anisotropy is scientifically valuable because real growth media often have preferred crystallographic directions. In this engine it is a designed angular bias, not a material crystal structure.

Injection rate currently lacks an independent steady-state effect

The control Injection rate scales the far-field Dirichlet value. Laplace’s equation is linear: multiplying all nonzero boundary values by a constant multiplies the relaxed field and every gradient by the same constant. After raising to (\eta), all candidate weights receive the common factor injectionRate^eta. The engine then normalises by the total weight before sampling, so that factor cancels.

Consequently, once the field has relaxed, Injection rate does not change the relative growth probabilities or the number of attempted sites. Growth per step and Sticking probability control event count. Injection rate can create transient differences immediately after a live change because the old potential remains in the grid, but it is not an independent steady-state kinetic control under the present normalised rule.

This is a P0 scientific-contract defect. The parameter should be removed, relabelled as a potential-scale diagnostic, or connected to an unnormalised event rate.

Painted “blocks” are grounded sinks, not neutral walls

A painted obstacle is excluded from occupation, but the potential solver sets it to (\phi=0), the same value as the aggregate. In a physical field problem that is a grounded or absorbing boundary. It modifies the harmonic field and can draw flux toward itself even though the cluster cannot occupy it.

An impermeable obstacle would more naturally impose a no-flux Neumann condition. The current brush target should therefore be called Grounded blocker or Field sink, or the boundary implementation should change to a reflecting/no-flux wall.

Directional bias is split across two different mechanisms

The brush target labelled Directional bias paints a scalar field. It increases growth weight in a region but stores no direction. The separate advanced Force/Bias drag records a global direction from pointer motion and favours gradients aligned with it.

The scalar brush should become Growth preference or Favoured zone. The drag tool can retain Directional bias. Conflating the two obscures whether the visitor is changing location or direction.

Laplacian moving-boundary problems connect several twentieth-century traditions.

Saffman and Taylor’s 1958 work on the displacement of a viscous fluid in a porous medium or Hele–Shaw cell established how an interface can become unstable and develop fingers. The pressure field in the fluid supplies a classic physical example of harmonic-field-driven boundary motion.

In 1981, Thomas Witten and Leonard Sander introduced diffusion-limited aggregation as a minimal stochastic model of particles diffusing and sticking to a cluster. It showed that a simple irreversible rule could generate ramified fractal aggregates.

In 1984, L. Niemeyer, L. Pietronero and H. J. Wiesmann introduced a stochastic dielectric breakdown model in which growth probability is controlled by a power of the local electric field. The exponent (\eta) provided a tunable bridge between compact and highly tip-selected structures.

Subsequent work unified many such systems under the heading Laplacian growth, including stochastic conformal-map formulations such as the Hastings–Levitov model. The common subject is not one material but a harmonic field coupled to a moving interface.

AL.Floys belongs to this computational lineage as an interactive lattice approximation. Its presets borrow visual vocabulary from several physical manifestations, but its actual scientific identity is the shared harmonic-growth mechanism.

Aspect	Current implementation
Entities / field	A binary occupied cluster on a rectangular grid, an age field, a scalar potential, an obstacle mask and a scalar preference field.
Potential equation	Approximate discrete Laplace equation solved with a limited number of four-neighbour Jacobi sweeps.
Boundary conditions	Aggregate and painted obstacles at zero potential; outer border or top edge at a positive far-field value.
Growth perimeter	Empty non-obstacle cells with at least one occupied cardinal neighbour.
Growth probability	A code-specific gradient proxy raised to (\eta), modified by compactness, anisotropy, painted preference, global directional bias and noise, then normalised.
Attachment	One or more candidates are sampled per frame and pass a separate sticking-probability gate. Growth is irreversible unless the visitor erases cells.
Source modes	Centre seed, bottom-border band with top source, or several random seeds.
Numerical approximation	Field is not fully solved at every frame and is not recomputed between multiple same-frame growth events.
Boundary geometry	Square grid, cardinal aggregate adjacency and optional angular fourfold or nominal sixfold bias.
Randomness	Seed placement, weighted candidate sampling, sticking gate and multiplicative growth noise.
Rendering interpretation	Occupied-cell age, palette, contrast, potential-like glow and optional trails. The images do not identify one physical material.
Public simple controls	Selectivity (\eta), Sticking probability, Anisotropy, Growth per step, Source mode and Palette.
Public simple tool	Seed only. Inject, grounded blocker, growth preference, directional bias and erase exist in the engine but are hidden from the simple surface.
Known contract issues	Injection rate largely cancels after normalisation; low-(\eta) help mislabels DLA; “surface tension,” “block” and painted “directional bias” overstate their implementations.

The most truthful one-line description is: a stochastic lattice boundary grows where an approximately harmonic field assigns the greatest relative flux.

Tip amplification — exposure becomes advantage

A branch that extends slightly beyond its neighbours encounters a larger field gradient. It receives more growth probability, extends again and turns a small fluctuation into a macroscopic lead.

Screening — the winner suppresses the interior

Deep fjords and branches behind a leading tip receive little harmonic measure. Their growth nearly stops even though they remain on the perimeter. The aggregate is organised by competition at a distance.

Branch splitting — no tip owns the future

Noise or field geometry can create two comparable high-flux sites near one tip. They separate, then compete, producing a fork whose later asymmetry records early random choices.

Source geometry — one equation, different global form

A centre seed grows radially toward an outer source. A border seed advances toward the top. Multiple seeds compete for the same field. Boundary conditions shape morphology as strongly as local rules.

Anisotropic needles — preferred directions channel growth

Strong square anisotropy can suppress off-axis branching and create long competing needles. Nominal hex bias can produce sixfold visual motifs, though the underlying grid remains square.

Compactness versus ramification

Lower selectivity and stronger neighbour smoothing distribute growth across more of the perimeter. Higher selectivity and weak smoothing concentrate it at exposed tips. The transition is a regime change, not simply “more branches.”

Competition among seeds

In a multi-seed world, one aggregate can screen another. The smaller cluster may stop not because it has exhausted material but because the shared field has been captured elsewhere.

First 30 seconds

1. Load Frost Garden and follow the newest branch tips rather than the completed interior.
2. Increase Selectivity (\eta) in stages. Watch the active perimeter narrow toward fewer tips.
3. Lower (\eta) below one. Look for broader growth and more frequent filling of side regions.
4. Load Needle Competition to see explicit angular preference.
5. Change Source mode from Centre seed to Border source and use Restart. Compare radial screening with a front advancing across the field.
6. Use Seed to add a competing nucleus near—but not touching—the existing aggregate. Watch how the two clusters divide the future field.

Three experiments

Experiment	Question	Do	Watch for	Why it matters
Field selectivity	How does one harmonic field produce compact and needle-like forms?	Same seed and source → compare (\eta\approx0.3), (1), and (3) with restart between runs.	Broad perimeter growth, DLA-like ramification near one, then extreme tip dominance.	Makes the exponent’s role visible and corrects the current low-(\eta) help.
Boundary conditions make worlds	Is the pattern determined only by local attachment?	Compare Centre seed, Border source and Multi-seed under otherwise similar settings.	Radial dendrites, advancing fronts and competitive screening.	Shows that the distant source is part of the causal model.
Symmetry imposed and broken	How does anisotropy compete with stochastic branching?	Set anisotropy near zero, then high Square, then high Hex, restarting from the same source mode.	Free branching, cardinal needles and sixfold-looking motifs.	Separates harmonic screening from designed directional preference.

A fourth advanced experiment uses the current Block brush. Paint a barrier beside the growing tip and observe that it acts as a zero-potential sink as well as an excluded region. That behaviour should be taught only after the target is relabelled.

Parameter	What it really controls	Increase it	Decrease it	What to watch
Selectivity (\eta)	Exponent applied to the field-gradient proxy before normalised sampling.	Concentrates growth on the highest-flux tips.	Flattens perimeter differences; (\eta\to0) approaches near-uniform perimeter selection.	Branch thickness, active-tip count and fjord freezing.
Sticking probability	Chance that a selected candidate actually becomes occupied.	More successful attachments per attempted event.	More skipped selections and slower growth.	Time rate, not the relative harmonic preference.
Growth per step	Number of weighted site selections attempted per animation frame.	Faster batched growth using one stale field solve.	More nearly sequential quasistatic growth.	Speed and same-frame branch thickening.
Potential iterations (advanced)	Jacobi relaxation sweeps before growth.	Field better adapts to the current cluster, at greater cost.	More lag and grid-transient artefact.	Stability of tip selection after rapid growth.
Anisotropy	Strength of a fourfold or sixfold angular multiplier.	Stronger preferred directions.	More isotropic stochastic branching.	Needles and apparent crystal symmetry.
Anisotropy mode (advanced)	None, square or nominal hex angular preference.	Not scalar.	Not scalar.	Remember that all modes still use a square grid.
Surface tension (advanced; relabel)	Neighbour-count compactness bias, not curvature physics.	Fills concave sites and smooths the front.	Leaves tip competition less regularised.	Front thickness and fjord closure.
Source mode	Initial aggregate geometry and far-field boundary placement.	Not scalar.	Not scalar.	Radial, planar-front or competitive growth.
Seed count (advanced)	Number of starting nuclei in multi-seed mode.	More competition for harmonic measure.	Simpler single-cluster growth.	Screening between aggregates.
Noise (advanced)	Multiplicative jitter on candidate weights.	More stochastic branch selection.	More deterministic response to field and anisotropy.	Fork symmetry and run-to-run variation.
Injection rate (advanced; defective)	Common scale of far-boundary potential.	Produces a transient field rescaling; after relaxation the common factor cancels in normalised probabilities.	Same.	It should not be taught as a steady growth-rate control.
Palette (visual)	Age/occupancy colour mapping.	Not scalar.	Not scalar.	No growth dynamics change.

The best public teaching controls are Selectivity, Source mode and Anisotropy. Sticking probability and Growth per step mainly alter the clock and numerical batching.

Frost Garden

Moderate selectivity and weak nominal hex anisotropy produce dendritic frost-like morphology. Keep the poetic title, but describe it as a frost-like harmonic aggregate, not a simulation of water vapour, latent heat or crystal thermodynamics.

Lightning Tree

High (\eta) strongly favours exposed tips and approaches the visual logic of dielectric-breakdown branching. It is the closest preset to the engine’s model lineage, yet it still lacks ionisation, charge carriers, dielectric strength and electric discharge time.

Viscous Crown

A border source and lower selectivity create a broad advancing front with fingers. Rename it Fingering Front or Border Crown. It is not a Saffman–Taylor fluid calculation and the current “viscous” adjective claims absent physics.

Needle Competition

Strong square anisotropy creates axis-aligned tips that compete for flux. This is one of the most scientifically legible presets because its title describes morphology and competition rather than a material identity.

Coral Front

Several seeds generate interacting branched forms. Coral is a visual analogy only; there are no organisms, accretion biology, nutrients or hydrodynamic environment.

Black Snowflake

High nominal hex anisotropy and compactness bias produce snowflake-like symmetry. The underlying neighbourhood remains square and the “surface tension” term is heuristic. Keep the aesthetic title with an explicit snowflake-like qualifier.

Recommended learning order: Frost Garden → Lightning Tree → Needle Competition → Border Crown / current Viscous Crown → Coral Front → Black Snowflake.

• Seed is the only public simple tool. It directly marks a local patch as occupied, adding a new aggregate or joining an existing one. This is an external nucleation intervention, not stochastic growth through the harmonic measure.
• Inject material directly occupies some cells probabilistically. It bypasses the field-weighted boundary rule and should be described as forced growth.
• Grounded blocker (currently Block growth) prevents occupation but sets the painted region to zero potential, thereby altering the field as an absorbing boundary.
• Growth preference (currently Directional bias brush) paints a scalar multiplier that favours a region but stores no direction.
• Directional bias drag stores one global vector and favours gradients aligned with the drag.
• Erase removes aggregate, obstacle and preference state.

The strongest public experiment is to add a second seed. It modifies the boundary-value problem and lets the visitor observe competition mediated through the field rather than direct collision.

The engine does not simulate frost formation. It contains no vapour diffusion coupled to heat, crystal attachment kinetics or latent heat.

It does not simulate lightning or dielectric discharge in physical units. There are no charges, ionisation thresholds, plasma channels or electric current.

It does not simulate viscous fingering as a two-fluid flow. There is no viscosity ratio, pressure-driven interface velocity, Darcy law, capillary pressure or conserved fluid volume.

It does not simulate coral biology or snow crystallography. Those preset names describe visual resemblance.

The field is only approximately relaxed. Multiple cells can grow from one stale field solve. High Growth per step changes numerical fidelity as well as speed.

The square lattice and cardinal adjacency influence morphology. Nominal hex anisotropy does not create a true hexagonal discretisation.

The current “surface tension” is a neighbour-count compactness bias. Painted blocks are grounded potential sinks, not neutral reflecting walls. Painted “directional bias” is scalar.

Injection rate is not an effective steady-state growth-rate control under normalised sampling.

The defensible conclusion is:

A harmonic field can mediate long-range competition along an advancing boundary, amplifying exposed tips and screening recessed regions into irreversible branched form.

Laplacian Growth adds a moving boundary governed by a field outside itself. Unlike Fractals, which classify a fixed mathematical plane, this world changes irreversibly. Unlike Particle Life, branches do not arise from pairwise agents. The aggregate reshapes the field, and the reshaped field decides where the aggregate can grow next—a feedback loop between form and opportunity.

Foundational and primary sources

1. P. G. Saffman and G. I. Taylor, “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid,” Proceedings of the Royal Society A 245, 312–329 (1958). DOI: 10.1098/rspa.1958.0085.
2. Thomas A. Witten Jr. and Leonard M. Sander, “Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon,” Physical Review Letters 47, 1400–1403 (1981). DOI: 10.1103/PhysRevLett.47.1400.
3. L. Niemeyer, L. Pietronero and H. J. Wiesmann, “Fractal Dimension of Dielectric Breakdown,” Physical Review Letters 52, 1033–1036 (1984). DOI: 10.1103/PhysRevLett.52.1033.
4. M. B. Hastings and L. S. Levitov, “Laplacian growth as one-dimensional turbulence,” Physica D 116, 244–252 (1998). DOI: 10.1016/S0167-2789(97)00244-3.

Modern scientific context

5. B. B. Mandelbrot and C. J. G. Evertsz, “The potential distribution around growing fractal clusters,” Physica A 177, 386–393 (1991). DOI: 10.1016/0378-4371(91)90177-E. Useful context for harmonic measure and screening.