Lichtenberg Figures: How Dielectric Breakdown Grows a Fractal

Branches that solve an equation as they grow

A Lichtenberg figure is the fractal, tree-like scar left by an electrical discharge through an insulating material — the pattern frozen into wood after a lightning strike, or the pattern burned into acrylic with a high-voltage transformer, or the treeing damage that eventually kills a stressed cable insulator. The shape is not decorative accident; it is the visible trace of dielectric breakdown, and it can be reproduced numerically by solving the same physics that grows the real thing.

Between the branches and everywhere else in the insulator, the electric potential φ obeys Laplace’s equation, ∇²φ = 0, subject to a fixed high potential at the discharge tip and a fixed low potential (often ground) at the far boundary. Laplace's equation says the potential at any point is simply the average of its neighbours — there are no sources or sinks in the gap, only geometry.

The dielectric breakdown model

The Dielectric Breakdown Model (DBM), introduced by Niemeyer, Pietronero and Wiesmann in 1984, turns that field into a growth rule. Start with a single charged seed cell. At every step, solve Laplace's equation on the grid to get the potential everywhere, then look at every empty cell adjacent to the growing structure. Each candidate cell is added with a probability proportional to the local field strength raised to a power η:

P(cell i grows) ∝ φ(i) ^ η
solve ∇²φ = 0  with φ = 1 on the tree, φ = 0 on the outer boundary
repeat: recompute φ, pick next growth cell by weighted probability, add it to the tree

This single exponent η controls the whole character of the pattern. At η = 0 every candidate cell is equally likely regardless of field strength, and the model reduces exactly to diffusion-limited aggregation (DLA), the same random-walk clustering process that builds soot particles and some mineral dendrites, giving loose, spindly, many-branched trees. As η increases toward 1, growth concentrates ever more strongly on cells where the field is already highest — the tips of existing branches, where the potential gradient piles up — which is exactly the physical feedback loop that makes lightning and Lichtenberg trees self-reinforcing: a branch that grows slightly ahead of its neighbours experiences a stronger field at its tip, so it keeps outrunning them, and thin, sparse, sharply branching main channels dominate over stubby side growth.

Why real Lichtenberg figures look the way they do

Feed a few thousand volts to a block of acrylic through a needle electrode and the same feedback plays out physically: free charge injected at the tip finds the path of least resistance is also the path of highest local field, ionises a thread of material microseconds ahead of the rest, and that thread becomes the new tip. The result is a main channel with self-similar side branches at many scales — a genuine fractal, with a measured dimension around 1.7 in two dimensions, close to the DLA fractal dimension because both processes are governed by the same harmonic, Laplacian growth law. On wood struck by real lightning, the discharge follows moisture paths through the grain rather than a uniform dielectric, which is why every branch pattern is unique even though the growth law is the same equation everywhere.

What this simulation lets you see

The simulation solves the Laplace field on a grid every few growth steps (a Gauss–Seidel or successive-over-relaxation solver is cheap enough to keep this interactive), grows one or a few cells according to the φ^η rule, and repeats. Sliding η from 0 toward 1 in real time is the fastest way to feel the difference between DLA's diffuse clustering and true dielectric-breakdown treeing — the same knob nature turns with material conductivity and applied voltage.

Frequently asked questions

Try it live

Everything above runs in your browser — open Lichtenberg Figures and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.