PLOTCASTER Photo to line art
Patterns

Free fractal SVG files, drawn from their rules

Free fractal SVG files, each drawn from its rule to a stated order: the Hilbert, Peano, dragon and Gosper curves, the Koch snowflake, the Sierpinski triangle and carpet, fractal trees, an Apollonian gasket and the Mandelbrot set. Nothing is traced, so every corner is where the mathematics puts it.

20 designs. Free for personal and commercial use, no sign-up. What you can do with them.

Or choose one below for its PNG, DXF and cutting sizes.

How each fractal is made

A fractal is a shape built from smaller copies of itself. The curves here come from a rewriting rule: start with a short string of moves, replace every letter by a longer string, and repeat. The number of repeats is the order, and each note gives it. The Hilbert curve of order 4 has 255 steps and order 5 has 1,023; the true curve is what you would reach if you never stopped. The carpets and trees are made by a shape that places smaller copies of itself, and the Apollonian gasket by filling every gap between three touching circles with the one circle that fits.

A file has to stop somewhere. Each design stops at the order where the smallest part is still wide enough to cut or draw at 100 mm, and the design page lists that width at seven sizes.

Plotting, engraving and cutting

The Hilbert, Moore, Peano, dragon, Gosper and arrowhead curves are each a single line, which is why pen plotter owners like them: one pen-down, no lifts. In these files the line has a width, so a plotter or laser follows both of its edges; for a single pass, engrave the shape as a fill. The carpet, the Vicsek fractal, the T-square, the trees and the gasket are single pieces and cut well from card, vinyl or thin plywood at 100 mm or larger. The solid Sierpinski triangle and the Cantor set are loose pieces by their nature, so print or engrave those.

How to open the files

  • Cricut Design Space: choose Upload, pick the SVG, and add it to the canvas. Set the width you want; the shape is already welded.
  • Silhouette Studio: Designer Edition and above open the SVG. The free Basic Edition does not, so open the DXF instead.
  • LightBurn, xTool Creative Space and Glowforge: import the SVG, then set the layer to cut, score or engrave.
  • Inkscape, Illustrator and Affinity: open the SVG. It is a plain filled path, so you can recolour it, add a stroke or combine it with your own work.
  • Printing or sublimation: use the PNG, which has a transparent background.

Each design page lists the file’s narrowest part at seven sizes, so you can tell whether it will survive at the size you plan to cut.

Questions

What does the order of a fractal mean?

How many times the rule has been applied. Order 1 of the Hilbert curve is three steps in the shape of a staple; each order after that replaces every cell with four smaller copies, so the step count runs 3, 15, 63, 255, 1,023. Higher orders look finer but their lines get too close together to cut.

Which fractal is best for a pen plotter?

A space-filling curve: the Hilbert, Moore, Peano or Gosper curve. Each is one continuous line with no crossings, so the pen never lifts and never passes over wet ink. The Moore curve is a closed loop, which also suits a cutter because the blade ends where it began.

Why does the Mandelbrot set here have no threads?

The true set has threads thinner than any line and tiny copies of itself strung along them. A cut file cannot hold those, so this one keeps the main body: the heart, the large disc and the buds on them, down to parts about a millimetre wide at 100 mm.

More free SVG files