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Lots of substitution tilings

Ed Pegg, Wolfram Research
Posted 7 years ago

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, φ, ψ, χ, ρ. 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. SqrtChi tiling SqrtRho tiling SqrtPsi tiling SqrtPhi tiling Quartic Pinwheel tiling SqrtTwo tiling PsiQuad Rho Quad tiling Trib Trap tiling Psi Trap tiling Psi Chord tiling Psi Wedge tiling Trib Chord tiling TwoTriangle tiling RhoQuad tiling Birds and Bees tiling Binary tiling Penrose Rhomb tiling Robinson tiling Kites and Darts tiling Ammann Chair or Scherer Golden Bee Ammann A4 tiling WaltonChair tiling Ammann Phi Chair tiling Triangle Duo tiling Equithirds tiling Tritan tiling limhex tiling Pinwheel tiling

Any corrections, suggestions or additions are welcome.

Attachments:
SubstitutionTili...nb
POSTED BY: Ed Pegg
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John Berglund
John Berglund
Posted 3 years ago

I'm not sure what you mean by "algebraic tilings." It doesn't seem to match the "Algebraic Tiling" article by S. K. Stein. I just saw your AlgebraicSubstitutionTiling function. ResourceFunction["AlgebraicSubstitutionTiling"]["KitesandDarts", 0] will give the substitutions used to define the "Kites and Darts" tiling. I wish I could define my own substitutions and see what the resulting tilings would look like. For example: A simple substitution tiling

This uses relatively simple shapes. It's got an inflation factor of sqrt(3). For a more complex example here is the beginnings of substitution tiling with regular 14-gons, 7-gons and other polygons that fit with them. 7-fold girih type tiling

As of now, I haven't computed it's inflation factor. How easy is it to program a new substitution tiling such as one of these? Thanks. John Berglund
POSTED BY: John Berglund
POSTED BY: Ed Pegg
John Berglund
John Berglund
Posted 3 years ago

I'm new to the community. The examples are great. How would you go about making up a substitution tiling of your own? It would be nice if you could specify what the substitutions would be for each shape, and have the program show what some patches would look like.

POSTED BY: John Berglund

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I've fixed a gap error in AlgebraicSubstitutionTiling that didn't handle complex substitutions well. Now it handles them fine. For example, the new Harvest tiling. Andrew Hudson's Harvest
Here's a few steps of it with my new code. harvest with new code

Here's the Dale Walton quintic from above. DaleWaltonQuintic

But it seems with the changes I've added some minor errors, so I need to work on it more. Then I need to add some complicated substitution tilings that went wonky before due to the gap error.

EDIT: Fixed all errors, I think. Let me know.

Attachments:
SubstitutionTili...nb
POSTED BY: Ed Pegg
Brad Klee
Brad Klee
Posted 7 years ago
POSTED BY: Brad Klee
POSTED BY: Ed Pegg
Brad Klee
Brad Klee
Posted 7 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 BY: Brad Klee

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 BY: Ed Pegg

So how to make these? Dale Walton sent me a picture of a new tiling. Walton 3017 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. Walton 3017 tiling

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:
Walton3017.nb
POSTED BY: Ed Pegg

May as well put in the rest of what I have so far.

Domino Tiling Drafter 3 tiling Drafter 4 tiling half hex tiling kite tiling l tromino tiling p pentomino tiling quarterhex tiling sphinx tiling tritan 4 tiling tromino tiling
POSTED BY: Ed Pegg
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