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In aperiodic domino tiling, the substitution/inflation methods (also named cut-and-project) seem to be the most commonly used. Here first sequence of a domino tiling described by this other question. It is generated by a fractal and its grid:

enter image description here

... Colored steps from this generator. enter image description here

Substitution rules

It is a square surface, so we need to start inflating 2 dominos (labeled or marked or colored prototiles), left and right:

Rules R1, [R1L] and [R1R]:

enter image description here

... and two symmetric rules for left-horizontal and right-horizontal dominoes:
Rules R2, [R2L] and [R2R]:

enter image description here

Using the rules to construct the tiling surface:

enter image description here

enter image description here

... Oops, the "?" shows a problem. We need more labels?
Here a a suggestion, adding labels "△" and "▽":

enter image description here

Are the rules correct? Where's the proof?

I don't know the correct term, "substitution/inflation rules" or "labeled rules" or "cut-and-project rules"... But the rules above are intuitive and illustrated. The problems are:

  1. Can I use labels (sides L/R) in the tiles?
  2. Are these four rules enough?
    How to proof that the rules are generating the tiles?
    Perhaps a modified Domino.py can generate. (the script generated this similar domino tile)
  3. The equivalence (between the fractal's grid and the domino surface) exists for any number N>3 of steps?

The main problem, 2, is the question here.

Notes

This is the "fractal version" of the domino, but I am supposing that we not need it to solve the problem.

enter image description here

Illustrating the original "fractal transform", from Hilbert curve and its grid:

enter image description here

Peter Krauss
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  • It's unclear what you're asking here. Also, isn't this a substitution/inflation rule method rather than cut-and-project? – Dan Rust Sep 03 '25 at 11:05
  • What's the question? – Toph Sep 03 '25 at 11:37
  • Hi @DanRust, I edited, see "The main problem, 2, is the question here." – Peter Krauss Sep 03 '25 at 12:41
  • Hi @Toph, I edited, see "The main problem, 2, is the question here." – Peter Krauss Sep 03 '25 at 12:41
  • Are these two rules enough for what? Also, the rules seem ambiguous, as the images appear to require an L or R label, as well as orientation labels (not just the starting tile, otherwise how do you iterate?). – Dan Rust Sep 03 '25 at 12:52
  • I think that might be the answer -- no, these rules aren't correct (or at least aren't complete), because they don't show how the dominoes in the dissection are oriented. If they did, it would be more obvious that the four different rules for "vertical left", "vertical right", "horizontal left" and "horizontal right" are the same but rotated and possibly mirrored. – Toph Sep 03 '25 at 13:19
  • @DanRust I edited, see "Using the rules to construct the tiling surface" – Peter Krauss Sep 03 '25 at 13:34
  • You still haven't said what type the image tiles are, so it's not clear how to iterate the rule. – Dan Rust Sep 03 '25 at 13:49
  • @DanRust, I edited again... Now you can explain your vision on your answer. – Peter Krauss Sep 03 '25 at 14:05
  • @Toph, I edited. Yes, need 4 rules. – Peter Krauss Sep 03 '25 at 14:06
  • @PeterKrauss This still doesn't show the orientation corrrectly. In this image: https://i.imgur.com/ROsk3PI.png I've highlighted one of the vertical dominoes in your Step 2 which is subdivided into four dominoes at Step 3... but it doesn't follow either rule R1L or R1R, as the resulting subdivision is upside-down. – Toph Sep 03 '25 at 14:23
  • Sorry @Toph, only now correcting, see edition and new clues. – Peter Krauss Sep 09 '25 at 03:12

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