The Rauzy Gasket

A shader I created for Pierre Arnoux, showing the Rauzy gasket.

The Rauzy gasket is a planar fractal. Its points are the admissible letter-frequency vectors of ternary episturmian words: triples $(u, v, w)$ with $u + v + w = 1$ recording how often each of three letters appears in a word whose complexity just barely exceeds the periodic threshold.

The simplex carries three renormalization maps. For a frequency vector with $u$ the largest coordinate, the map

$$f_x(u, v, w) = \frac{(u - v - w,\; v,\; w)}{u - v - w}$$

erases the letter following each non-$x$ letter and renormalizes. The other two maps $f_y$ and $f_z$ act symmetrically. Applied repeatedly, these maps drive any interior point toward the boundary; the gasket is the invariant set where this process never terminates.

The shader iterates all three maps in turn, at each step choosing whichever keeps the point inside the simplex (all three barycentric coordinates positive). After 10,000 steps the colouring encodes the value as the point escapes, darkening the longer it goes.

Pierre Arnoux and Štěpán Starosta “The Rauzy Gasket” (Further Developments in Fractals and Related Fields, Birkhäuser, 2013, pp. 1–22)