Rauzy Fractal

Explore how substitutions and eigenvectors produce self-similar geometric fractals.

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Question

How does a symbolic substitution create a self-similar geometric fractal?

A three-step diagram showing a substitution word, its prefix-count lattice path, and projection to a Rauzy fractal.
Method: iterate a symbolic substitution, count prefixes as a lattice path, then remove the expanding direction by projecting onto the contracting plane.
A Rauzy fractal separated into three coloured pieces, each resembling a transformed copy of the whole.
Structure: separating points by their final symbol reveals smaller transformed pieces that fit back into the complete fractal.
A two-dimensional Rauzy fractal divided into three coloured pieces.
A two-dimensional Rauzy fractal computed in the project repository.
A three-dimensional Rauzy fractal divided into three coloured pieces.
A three-dimensional projection produced from the same symbolic substitution.

Goals

  • Construct a Rauzy fractal from a Pisot substitution.
  • Study its projection, self-similarity, and two- or three-dimensional form.

Method

Turn an expanding symbolic sequence into a lattice path, then project away its main growth direction.

  • Choose a substitution and compute its substitution matrix and eigenvectors.
  • Iterate the substitution and record the prefix-count path.
  • Project the path into the contracting space and compare the resulting pieces under substitution.