The Rauzy fractal is a plane-tiling figure which is composed of 3 copies of itself, of different sizes, with a constant ratio between the areas of the various copies. It was discovered by Rauzy in 1982.
It takes a little thought to work out an IFS which generates this figure.
Assuming that the ratio of areas between elements is indeed constant, and denoting the ratio of areas as a, we have a³+a²+a=1. Solving this numerically we get a~=0.543689012692. The linear contraction, which we denote c is the square root of this. Thus we have a linear contraction of c for the 1st element, of c2 for the 2nd element, and of c.3 for the 3rd element.
We still have to ascertain the orientations and positions of the 3 elements (giving 9 unknowns).
The following image shows the fractal with the 1st element redivided into 3.
Note that the red and green parts have the same orientation, as do the blue and magenta parts. Note also that the displacement between the members of these two pairs is the same. Let us denote the transform corresponding to the 1st element as T1. Note that the red part is the image of the 1st element under T1, so we can deduce that the transform corresponding to the 2nd element, denoted T2, is T1.T1+x where x is a vector. Similarly the blue part is the image of the 2nd element under T1, so the transform corresponding to the 3rd element, denoted T3, is T2.T1+x. So we have
The magnitude of x only affects the size of the attractor, and the orientation of x only affects the orientation of the attractor, so we can arbitrarily fix this as the vector (1,0). We are left to ascertain the nature of T1. This consists of a c-fold contraction and a rotation, so we are left to ascertain the angle of rotation. Denote this as a. If we consider the 5 transforms (as for the previous image)
and plot the attractor corresponding to the 25-element IFS that is the cartesian product of this set
we can identify a triangle from which it can be deduced that
Modifying T1 by adding a vector (-1,0) has the effect of changing the position of the attractor. This was found convenient as it results in the coordinates of the first two triple junctions being (-1,0) and (0,0) and simplifies drawing the guide lines shown above.
Solving this numerically gives a value for the angle of 235.311°. Plotting an IFS with these values of c and a gives a result which is accurate at a resolution of 12000×12000.
Use of an angle of 360°-235.311° results in the mirror image.
The Rauzy fractal tiles the plane with a single copy in the unit cell.
The tiling shown above is obtained by rotating each copy by 3a. With the construction given above the tiling vectors are (2.5,0) and (2.5cosa,2.5sina).
Examination of the 5 element rendition shows that Rauzy fractal also tiles the plane with a two copies in the unit cell.
There are also aperiodic tilings.
There is alternative, symmetrical, 5 element version of the Rauzy tile (on the right). This is created from the one on the left by adding to the vector (ccosa, csina) to the transform corresponding to the red element, the vector (c².cos(2a)-ccosa, c².cos(2a)-csina) to the transform corresponding to the blue element, and the vector (1,0) to the transform corresponding to the gold element.
There are two, symmetrical, 9 element versions. (Using copies of 3 sizes; others can be created if 4 sizes are used.)
From any of these symmetrical implementation of the Rauzy tile demiRauzy fractals can be mechanically derived.
MetaRauzy fractals can be mechanically derived from any of the 3, 5 or 9 element implementations of the Rauzy tile, or indeed any other implementation.
Related Fractals: demiRauzy fractals
Source: Reverse engineered from an image published by Anne Siegel of the Institut de Mathématiques de Luminy (University of Marseilles).
References
© 2001 Stewart R. Hinsley