A two part triangle space fill that isn't one of the Kenyon 17: http://www.math.brown.edu/~rkenyon/gallery/all17.pdf
As you know I spent some time making Mathematica code for all 17
tiles. The triangle space-fill in the Kenyon 17 is symmetrical
and is actually related to the von Koch set.
This affine with the 1 to Sqrt[6] affine scaling is new. ( at least to me). The result was found experimentally in the area of simplex affine triangle IFS’s.
Steward Robert Hinsley would be happy. https://lh3.googleusercontent.com/-B0J8eYkg750/WIDyOVnIl0I/AAAAAAABFO8/gkf6B8b-HX8KGv9oSbZ5botXPRhZpBqJgCL0B/w506-h365/.10_affine_triaqngle_spacefill3000000_crs2.png
Broken Rule Tiling:
Tiling with self-similar overlap:
broken rule tiling;
This tiling is two transforms
and not any kind that Kenyon ever contemplated in his 17.
(*Mathematica program*)
Clear[f, dlst, pt, cr, ptlst, M, r, p, rotate]
allColors = ColorData["Legacy"][[3, 1]];
firstCols = {"Red", "Blue", "Magenta", "Green", "DarkOrchid", "LightSalmon",
"LightPink", "Sienna", "Green", "Mint", "DarkSlateGray", "ManganeseBlue",
"SlateGray", "DarkOrange", "MistyRose", "DeepNaplesYellow", "GoldOchre",
"SapGreen", "Yellow"};
cols = ColorData["Legacy", #] & /@
Join[firstCols, Complement[allColors, firstCols]];
Length[cols];
(*IFS broken rule fractal tiling by Roger L.Bagula 19 Feb 2018©*)
(*IFS \
program type*)
{m1,
m2} = {{{0.7414178834516808`, 0}, {0,
0.7414178834516808`}}, {{0.671043606703789`, 0}, {0,
0.671043606703789`}}};
A = {{0, 1}, {-1, 0}}
a1 = Expand[(A.m1.{x, y}).A.m1.{x, y}]
a2 = Expand[(A.m2.{-x, -y}).(A.m2.{-x, -y})]
sc = Sqrt[1/FullSimplify[ExpandAll[(a1 + a2)/(x^2 + y^2)]]]
f[1, {x_, y_}] = Expand[A.m1.{x, y} + {-1, 0}];
f[2, {x_, y_}] = Expand[A.m2.{-x, -y} + {1, 0}];
pt = {0.5, 0.5};
{p1, p2} = {Det[sc*A.m1], Det[sc*A.m2]}
dlst = Table[ Which[(r = Random[]) <= p1, 1, r <= 1, 2]
, {n, 3000000}];
cr[n_] := cr[n] = cols[[n ]];
{0.5497004779019702`, 0.45029952209802954`}
ptlst = Point[
Developer`ToPackedArray[Table[pt = f[dlst[[j]], pt], {j, Length[dlst]}]],
VertexColors -> Developer`ToPackedArray[cr /@ dlst]]; Graphics[{PointSize[.005], ptlst}, AspectRatio -> 1, ImageSize -> 1000]
(*end program*)
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