Curve of fractal triangle.

Question

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!

Answer 1

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.

Answer 2

ifstile.com

example: x^3+x-1.aifs

set: 1.029524_19

Just click boundary mode in the toolbar and then open console.

You can print different information about the boundary: definition, measure, dimension, diameters...

For example:

Base: x^3+x-1

used roots:

-0.3411639019140096+1.161541399997252*i

-0.3411639019140096-1.161541399997252*i

p = |prod(used roots)| ~= 1.465571231876768

dim = 2*log(x)/log(p)

dim1 ~= 1.029524059929087

Graph: x^7-2x^2-1

x ~= 1.217458216550893