# What Method is used for Projecting the Rauzy Fractal?

I am trying to construct the Rauzy Fractal (http://en.wikipedia.org/wiki/Rauzy_fractal), I have a Tribonacci word generator and have the stairs constructed but I can't seem to get the projection onto a 2D plane correct.

On the 4th slide of this: http://kmwww.fjfi.cvut.cz/jn08/slides/Thuswaldner.pdf they depict a bounding cube, my assumption was that I would project each point (x,y,z) onto (x',y') along the line (xMax, yMax, zMax)->(0,0,0). But they mention a Contracting Plane what does this terminology mean? Are all points projected along the same line or each to their own?

For this I am using the unit cube orthogonal projection matrix shown on the wiki: http://en.wikipedia.org/wiki/Orthographic_projection with Ortho(-1, 1, -1, 1, 1, -1) being my cube and (x/xMax, y/yMax, z/zMax) being my normalised point to be projected. I think I've got this part completely wrong. The result would agree...

Should I even be using the projection matrix?

Have a look at this paper by Sirvent and Wang. It's not elementary but it defines the process quite completely with a worked example. In particular, the contracting space $${\mathbb H}_c$$ is clearly defined. On slide 3 of the presentation, it simply says that $${\mathbb H}_c$$ is "generated by the eigenvectors of $$\beta$$ Galois conjugates". (Of course, it's a presentation so we could perhaps excuse the terseness.) The relevant part in the paper is the so called valuation map $$E$$ which accepts strings and returns complex numbers. The image of the orbit of the fixed word is exactly the Rauzy fractal.

Suppose, for example, that you're dealing with the substitution $$1\to 12, \: 2\to 13, \: 3\to 1.$$ This has incidence matrix $$\left( \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 0 \\ \end{array} \right)$$ one of whose eigenvectors is approximately $$\omega = \langle \omega_1, \omega_2, \omega_3 \rangle =\langle -0.412 + 0.61 i, 0.223 - 1.12i, 1\rangle.$$ The complex conjugate of $$\omega$$ is also an eigenvector; the remaining eigenvector corresponds to a real eigenvalue and not relevant.

Given a finite word $$U$$ let $$|U|_i$$ denote the number of occurrences of the symbol $$i$$ in $$U$$. Then, according to equation 3 in the paper, the valuation $$E$$ is defined by $$E(U) = \sum_{i=1}^3 |U|_i \omega_i.$$ Note that the $$\omega_i$$s are simply the components of the vector $$\omega$$ and have complex values. So clearly, $$E(U)$$ is a complex number.

Now, the fixed word for this substitution starts something like so:

12131211213121213121121312131211213121213121121312112131212131211213121312112131212131


Of course, he first few terms of the orbit of this finite word under the shift operator are

12131211213121213121121312131211213121213121121312112131212131211213121312112131212131
2131211213121213121121312131211213121213121121312112131212131211213121312112131212131
131211213121213121121312131211213121213121121312112131212131211213121312112131212131
31211213121213121121312131211213121213121121312112131212131211213121312112131212131


If we apply the valuation map to each of these, we obtain points in the complex plane. If we start with a much longer approximation to the fixed word and perform the process for many points in the orbit, we get a good approximation to the rauzy fractal. We even obtain a decomposition of the rauzy fractal by examining those terms in the orbit starting with 1, 2, or 3.

That is exactly the process I used to generate the following: