Curve of fractal triangle. Here's a triangle related to the Hokkaido tiling by Shigeki Akiyama.  I believe Stewart Hinsley found this particular tiling.  

The outer fractal triangle is divided into two similar fractal triangles.  
Is there a good generator for the fractal curves of the boundary?  
EDIT:  With help from William R. Somsky, Dieter Steemann, and Дмитрий Мехонцев, I managed to boil the boundary down to the following code.
r = Root[-1 - #1^2 + #1^3 &, 3];
init1 = { -1, r, r^3, - r^4};
init2 = {-r^6, -r^4, -1, -(r^3 + r^7)};
cowed[comp_] := First /@ Split[Flatten[RootReduce[#[[1]] + 
 (#[[2]] - #[[1]]) {0, -r^5, r^5 + 1, 1}] & /@ 
 Partition[comp, 2, 1, 1], 1]];
Graphics[{EdgeForm[Black], Green, 
 Polygon[ReIm[Nest[cowed[#] &, init1, 5]]], Blue, 
 Polygon[ReIm[Nest[cowed[#] &, init2, 5]]]}]


Behold the fractal cow-nautilus!


I've been able to find an IFS to generate the tile edges (rather than the interiors) producing graphs such as this (color fill added in post-processing):

I'll expand on this answer once I figure out the best way to present my results.
