Fractal chevron

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I started with a chevron consisting of four equilateral triangles.  Each of those was then subdivided into nine smaller triangles, so that alternating bumps and notches could be applied to the centre of each side on the perimeter (for practical reasons I used a reverse approach, scaling up sides by a factor of three and reducing once done).  In theory, this process can continue indefinitely to produce complex fractal-like outlines.  



There are left and right versions of the modified chevron, depending on which face is upward.  This means you cannot use both versions in the same tessellation.  Nine copies of each newly generated tile, can be put together to produce the next iteration.  Applying bumps and notches to even numbered polyiamonds, alters the tiling properties after the first iteration.

There are four permutations for the chevron but only two for the modified versions, i.e., first and third from the left.




These bumped and notched chevrons can be put together in six different ways (shown here as the second iteration).  Two of them are (modified) comets.





I made a symmetric board to fit fifteen of these shapes.  According to Jaap's Polyform Puzzle Solver, there are 128 solutions but this does not take into account duplicates (rotations).  Burr-Tools found 62 solutions with rotation and mirror symmetries off.  A little confusing, as the two results don't seem to marry up.

Below a few configurations.  I chose vertices of 6,3, 5 and 4 that meet at the centre.

















Comments

  1. James5 October 2023 at 13:16

    I recently found a similar thing with the chevron and comet but with a different edge modification. Replacing the edge with three segments separated by two 120 degree angles. It seems like we've been looking at similar things.
    Also for the chair tile: https://ibb.co/2yTmK8n - it would be interesting to know what tiling of the plane looks like for the chevron

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  2. Dave S.24 October 2023 at 13:59

    Only just seen this. Thanks for the chair tile link - looks really interesting. Where do I find your chevron and comet version?

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  3. James22 November 2023 at 00:55

    I also only just saw your reply yesterday. This is what I was thinking of https://ibb.co/zZLJRf6 . Hope you can make sense of these - they were just drawn out on paper. While making a digital version, I realised there's a simpler fractal possible. I've put that here: https://mathstodon.xyz/@jsmith/111450939694859453

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