As JA pointed out, at the end of https://en.wikipedia.org/wiki/Rauzy_fractal they say "For any unimodular substitution of Pisot type (??meaning what??), which verifies a coincidence condition (apparently always verified), one can construct a similar set called "Rauzy fractal of the map". They all display self-similarity and generate, for the examples below, a periodic tiling of the plane." Then they give 4 pictures, which you can click on to see enlargements. These pictures seem rather unconvincing to me, by the way. My own alleged tiling ("example #1") based on the "plastic number" 1.3247179572... presumably is of that ilk? Google then suggested that WP Thurston found my plastic-number-based tiling, or something like it, in the 1980s. It also found this paper: http://math.tsukuba.ac.jp/~akiyama/papers/Min_pisot.pdf which in fig 1 and fig 2 page 5 gives pictures of a tile arising from the "plastic number." Their allowed bitstrings forbid 11, 101, 1001, and 10001 as substrings? This sure does not look like it tiles plane by translation in a periodic way, but it does tile nonperiodically via 3 tile-scalings. They claim the Haussdorf dimension of the boundary of this tile is 1.10026... Wikipedia also claims the Rauzy tile will "tile the plane periodically by translation" in addition to the nonperiodic tiling I had in mind, involving 3 tile scalings and both translation & rotation. You can see pictures of both kinds of Rauzy tilings here: http://www.cant.ulg.ac.be/cant2009/lec-as1.pdf The "cut and project" method of generating a quasicrystal is, 1. start with some point-lattice in a higher dimension. 2. make a plane, or anyhow affine subspace, which is "irrationally oriented" with respect to the lattice directions. 3. Consider the lattice points nearby (e.g. within distance K of) that subspace. 4. project them orthogonally into the subspace. 5. result is a point set in your desired dimension, which is aperiodic, obeys point-density upper and lower bounds, but acts periodic in many approximate ways, e.g. fourier transform. This is simplest to think about for 1-dimensional quasicrystal arising from 2-diml lattice. I have heard claims that every quasicrystal can be regarded as arising in this way. To the extent that is true, "quasicrystallography" is the same thing as "crystallography with higher dimensions permitted." This view seems to have been adopted by at least one international society for crystallographic froobozzles. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)