# Lots of Substitution Tilings

Posted 2 years ago
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 I recently updated Substitution Tilings, one of my many items at the Wolfram Demonstrations Project. Some of these were introduced in my blog Shattering the Plane with Twelve New Substitution Tilings Using 2, ?, ?, ?, ?. Here are 26 of the 40 tiling currently in Substitution Tilings. Some but not all of these are at the Tilings Encyclopedia.Wolfram Language code is attached below at the end of this post. Any corrections, suggestions or additions are welcome. Attachments:
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Posted 2 years ago
 May as well put in the rest of what I have so far.
Posted 2 years ago
 So how to make these? Dale Walton sent me a picture of a new tiling. "All edges are powers of x=1.2365057033915... (5,6,7) triangle divides into (0,5,6); (3,4,5); (2,4,5)" Where $x^5-x^3-1=0$. This particular root has discriminant 3017. It's an algebraic number field seen giving extremal solutions in Wheels of Powered Triangles and Degenerate Power Simplices.By the end of the notebook, I get to this image. Not quite there. For a full substitution tiling system there should eventually be a fixed number of colors where every color represents congruent triangles. I haven't solved that yet for this tiling. Attachments:
Posted 2 years ago
 I should mention the Demonstrations of Dieter Steemann and the tiling demos of Karl Scherer, particularly Rep-tiles and Irreptiles. I'm still gleaning tilings from these, the Tilings Encyclopedia and IFStile. Hopefully I'll be able to improve my Demonstration Substitution Tilings to be stronger with a lot more tiling systems so that all of them are easily investigated.
Posted 2 years ago
 Hi Ed,Good that you got integer inflation factor tilings in the second post, but there are trivial examples Missingone square to four, or one equilateral triangle to four.You might also want to include 3D ABCK tiling by Danzer & Co. Ive already done some exploration of integer coordinatization, see for example:http://demonstrations.wolfram.com/TransformationOfIcosahedralSolidsInZ15/which I think you probably published some time earlier. I still can appreciate the result of this demonstration, and think that it suggests more to be done on your program here. If you decompose tiles to edges, how many unique vectors do you get? How are those vectors written out in a canonical basis?Cheers Brad
Posted 2 years ago
 Missingone square to four, or one equilateral triangle to four. How about one square to five? AlgebraicSubstitutionTiling[{1,{{-3,-1},{-3,1},{-1,-3},{-1,-1},{-1,1},{-1,3},{1,-3},{1,-1},{1,1},{1,3},{3,-1},{3,1}}, {{1,6,12,7}-> {{1,2,5,4},{5,6,10,9},{8,9,12,11},{3,4,8,7},{4,5,9,8}}} , {{1,1,1,1,1}}},5,{"N", "ImageSize"->{600,Automatic}}] Here's another one I was just looking at AlgebraicSubstitutionTiling[{Root[-1-#1^2+#1^3&,1],{{{4,0,0},{0,0,0}},{{8,-4,-1},{0,-8,7}},{{0,0,-4},{0,0,0}},{{0,0,0},{0,0,0}},{{-2,-3,3},{-6,3,1}},{{2,1,-1},{-6,3,1}},{{4,5,-6},{4,1,-2}},{{0,-3,2},{4,1,-2}}}/4, {{1,2,3}-> {{4,7,3},{1,6,4},{6,5,2},{7,8,4},{4,5,6},{2,8,7},{8,5,4},{5,8,2}}, {1,2,3}-> {{1,2,3}},{1,2,3}-> {{1,2,3}},{1,2,3}-> {{1,2,3}},{1,2,3}-> {{1,2,3}},{1,2,3}-> {{1,2,3}},{1,2,3}-> {{1,2,3}}} , {{0,2,3,3,4,4,5,5}+1,{1},{2},{3},{4},{5},{6}}},1,{"N", "ImageSize"->{600,Automatic}}] It's in the supergolden ratio, psi. But if you scale the area of the big triangle to the component triangles, you get areas psi^{7.7217, 5, 3, 2, 2, 1, 1, 0, 0} ... the big triangle is out of phase for a smooth substitution tiling system with a fixed number of sizes at each step.