Here are the (unique) fundamental tiling polygons for the Rauzy fractal. The fractal borders are shown in black, and the corresponding polygon borders are shown in red. Click "prev" or "next" to step the substitution backward or forward: https://www.karzes.com/xfract/img/rauzy.html?depth=1 Here's a color version showing just the polygons (you can switch between views by selecting "mode"): https://www.karzes.com/xfract/img/rauzy.html?mode=3&depth=1 These correspond exactly with the fractal tiles, in that the tilings have the exact same vertices, and as such they are unique. At each stage of the substitution, there are three sizes of polygons. The two smallest polygons have the same shape but different sizes, while the largest polygon has a different shape and size. Similarly, here are the unique tiling polygons for the two Penrose fractals (Hausdorff dimension ~1.266340), which almost no one knows about for some reason. Again, click "prev" or "next" to step the substitution backward or forward, and select "mode" to change the view: Penrose 1: https://www.karzes.com/xfract/img/penrose1_a1_set.html Penrose 2: https://www.karzes.com/xfract/img/penrose2_a1_set.html In the fractal case, these are exact substitutions, i.e. tile borders are not changed by the substitution (unlike the polygonal case). The only exact polygon substitutions I know of for Penrose tilings are the ones based on the component triangles, but those have 4 tiles instead of 2 (counting the refelected tiles as distinct from the un-reflected tiles): https://www.karzes.com/xfract/img/pfc1_13_set.html The better-known 2-tile polygonal Penrose tilings are formed by gluing adjacent reflected pairs of triangles together, but the resulting substitutions are no longer exact. Tom ed pegg writes:
At https://demonstrations.wolfram.com/SubstitutionTilings/ I was able to convert several Rauzy fractals into simple single polygon substitution tilings that had never been seen before. I'm convinced there is a nice 3D substitution tiling based on a single polyhedron that is currently unknown.
--Ed Pegg Jr
On Saturday, March 14, 2020, 06:01:38 PM CDT, Tom Karzes <karzes@sonic.net> wrote:
Right, I use the magenta color to indicate overlap. When the components fit perfectly to form solid tiles, this only occurs at their perimeters, but for the intermediate stages there is lots of overlap.
Tom
Fred Lunnon writes:
Ah, I see --- magenta indicates overlapping!
https://en.wikipedia.org/wiki/Rauzy_fractal
On 3/14/20, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Looks nice and runs fine for me on Firefox.
But the Rauzy stages have only 3 tile colours; what purpose is served by the magenta regions?
WFL
On 3/14/20, Tom Karzes <karzes@sonic.net> wrote:
Here's a GIF animation that shows the Rauzy fractal converging from an IFS by varying the angles in unison:
https://www.karzes.com/anim_rauzy.gif
Note that it converges at two distinct points in the sequence, in two different ways, so don't close it prematurely. At one point it almost looks like it's going to converge to a Heighway dragon, but of course it doesn't.
Tom