Lichtenberg Figure

Make a simulated Lichtenberg-style figure: a branching pattern grown on a grid by the dielectric breakdown model. The figure starts from a point, a short line or the bottom edge; the electric field round it is worked out, and each new cell is added next to the figure with a chance that follows the field raised to the power η. Low η gives bushy growth; higher η gives fewer, longer branches. The line width can follow each branch’s Horton–Strahler order, so the trunk is thickest and the tips are thinnest, on white or a dark background. Change the seed for a new pattern, then download a PDF, a 300 DPI PNG or an SVG with every branch as a vector line. Nothing is uploaded.

100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.

How to make a simulated Lichtenberg figure

1. Pick a preset, or choose where the figure starts.
2. Set the grid size, the number of cells, η and the seed.
3. Choose the line width (by branch order or even), the colours and the background.
4. Choose Fitted to one page or Real size and the paper.
5. Download the PDF and print at 100% (Actual size), or the PNG or SVG.

What you can use it for

• Posters and wall art of branching patterns
• Physics and maths lessons on growth models
• Plotter, laser-engraving and vinyl line art from the SVG
• Backgrounds for cards, covers and album art
• Tattoo and illustration line studies
• Comparing η values side by side

The 14 presets

Preset	Start	Grid	η	Cells grown	Stopped at	Tips	Highest order
From a point, η = 1, up to 900 cells	point	101 × 101	1	899	cell count	429	5
From a point, η = 2 (fewer, longer branches)	point	101 × 101	2	368	reached the electrode	178	4
From a point, η = 3 (sparse)	point	101 × 101	3	211	reached the electrode	65	4
From a point, η = 0.5 (bushy)	point	101 × 101	0.5	1,199	cell count	596	6
From a point, η = 0 (every edge cell equally likely: a compact blob)	point	101 × 101	0	899	cell count	498	5
Light branches on a dark background, η = 1.5	point	101 × 101	1.5	486	reached the electrode	238	5
From a short line, η = 1	line (21 cells)	101 × 101	1	1,011	reached the electrode	486	4
Up from the bottom edge, η = 1	bottom edge	101 × 101	1	999	cell count	516	4
Up from the bottom edge, η = 2	bottom edge	101 × 101	2	599	cell count	287	4
Fine grid 141 × 141, η = 1.5, up to 1,500 cells	point	141 × 141	1.5	1,112	reached the electrode	513	5
Coarse grid 61 × 61, η = 1, thick lines	point	61 × 61	1	295	reached the electrode	135	4
Even line width (no branch order), η = 1	point	101 × 101	1	899	cell count	423	5
Seed 42, η = 1.2, dark background	point	101 × 101	1.2	623	reached the electrode	294	4
Real size: 90 mm circle, η = 1.5	point	81 × 81	1.5	436	reached the electrode	198	4

Every count is worked out by the tool from the figure it draws.

How η changes the figure (point start, 101 × 101, seed 1)

η	Cells grown	Stopped at	Tips	Tips per cell	Highest order	Largest Laplace error
0	899	cell count	498	55.4%	5	2.4e−4
0.5	899	cell count	443	49.3%	5	2.4e−4
1	899	cell count	429	47.7%	5	2.4e−4
1.5	522	reached the electrode	235	45%	4	2.4e−4
2	368	reached the electrode	178	48.4%	4	2.4e−4
3	211	reached the electrode	65	30.8%	4	2.4e−4
4	149	reached the electrode	47	31.5%	3	2.4e−4

Each figure is grown to at most 900 cells or until it touches the outer electrode.

Branch orders in the first preset

Horton–Strahler order	Cells of that order
1	429
2	245
3	117
4	98
5	10

Order 1 cells are the tips and the thinnest lines.

What prints

Download	What is in it
PDF	The figure fitted to one page or at real size, vector lines, with the counts and a check bar
PNG 300 DPI	The same sheet as a picture
SVG	Every branch at the printed size as vector lines, one path per branch order

For a cutter that needs DXF, use SVG to DXF.

Which tool to use

You want	Use
Branching patterns grown by a field simulation	Lichtenberg Figure (this page)
The Lorenz equations solved and drawn	Lorenz Attractor
Cell patterns from scattered points	Voronoi Generator
Mandelbrot, Julia and line fractals	Fractal Generator
Soft organic shapes	Blob Generator

Other tools: the Voronoi Generator draws cell patterns, the Fractal Generator renders fractals and the Blob Generator makes organic shapes.

Frequently asked questions

What is a Lichtenberg figure?

A branching pattern left by an electrical discharge in or on an insulating material, named after the German physicist Georg Christoph Lichtenberg, who studied them in 1777. This page draws a simulated Lichtenberg-style pattern on a grid; it is a model drawn by our code, not a record of a real discharge.

How is the pattern grown?

With the dielectric breakdown model of Niemeyer, Pietronero and Wiesmann (Physical Review Letters, 1984). The figure is held at potential 0 and the outer electrode at 1; the potential of every other cell is solved so that each cell is the average of its four neighbours (Laplace’s equation on the grid). Every empty cell touching the figure is a candidate, chosen with a chance proportional to its potential to the power η, and joined to one neighbouring cell of the figure. The field is solved again after every new cell, until the largest error is below 0.001, which our tests check.

What does η do?

η sets how strongly the field picks the cells near the tips. With η = 0 every candidate is equally likely and the figure grows as a compact blob; as η rises the figure has fewer tips and reaches the outer electrode with fewer cells, so the branches look longer and sparser. The table on this page shows it for one seed, worked out from the grown figures.

Is it the same as diffusion-limited aggregation?

No. Diffusion-limited aggregation, proposed by Witten and Sander in 1981, grows a cluster from randomly walking particles; this tool does not walk particles, it solves the field and grows by the dielectric breakdown rule.

What is the Horton–Strahler order?

A number for how big a branch is, from river studies by Horton (1945) and Strahler (1952): every tip is order 1, and where two branches of the same order meet, the branch below has the next order up. The tool works it out for every cell, and Line width by branch order draws higher orders thicker.

Is the figure always one tree?

Yes. Every new cell is joined to exactly one cell already in the figure, so the branches never close a loop and all of them lead back to the start. Our tests check that for every preset.

Why did it stop before the cell count?

Growth stops when a branch touches the outer electrode, as the info line says; choose a bigger grid or a higher η to grow further before that.

How big can it go?

Grids from 41 × 41 to 141 × 141 cells and up to 2,000 cells; every preset draws in under 2 seconds in our tests.

Can it print at real size?

Yes, when the drawing fits on one sheet: choose Real size and set the printed width. Every printed point is within 0.2 mm of its true place in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar in the footer lets you confirm the scale. Bigger drawings print fitted to the page.

Is it free?

Yes. There is no sign-up, no watermark and no limit.

Is anything uploaded?

No. Every point and branch is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.