Anyone interested in tiling and science fiction should absolutely read Anathem by Neal Stephenson, it’s an excellent book that does an unexpectedly deep dive into tiling mathematics.
I finished re-reading Anathem last month. Unfortunately, AFAICT, the tiling problem depicted in the book is fictional. Though I keep wondering if it is possible to make a real puzzle conform to the (somewhat vaguely) specified problem in the book.
I invented a game that uses tiles in the shape of a regular hexagon plus a rhombus. IMHO, you can tile the plane aperiodically with them. See the first picture in [1].
I have been fascinated by regular and irregular tilings for a long time.
Last year, I spend some time generating irregular/random tiling made out of squares and triangles. One of the blogs about it is: https://www.iwriteiam.nl/D2208.html#23b
Interesting! I like your triangle square pattern as well. I haven't seen that before. You may already know that in the landscape architecture world your versaille-like tilings are frequently called ashlar patterns. I think they typically use periodic tilings. "AutoCAD ashlar drafting patterns" are available all over the place.
Apparently M. C. Escher liked the snub square tiling. Well it says that on https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling (which is dual to the snub square) according to the ref on wikpedia which is " "Dodecahedron", M. C. Escher Kaleidocycles".
The original monotile paper https://arxiv.org/abs/2303.10798 mentions that (at the time that was written) the minimum number of sides had to be 5 <= n <= 13, but also that a smaller tile can't be a polykite like the hat; exhaustive search had shown that there were no other aperiodic monotiles (than the hat and turtle) with < 21 kites
But a correction: the basic shape of the spectre is 14 sides with 2 of them colinear. It's only truly aperiodic when you modify the 14 edges, making it clear it is 'really' 14 sided not 13.
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[ 3.3 ms ] story [ 43.6 ms ] threadThis sort of “HN coincidence” has happened to me several times this week, is there a term for it I don’t know?
putting a hole in the center of a tiling grid turns a trivial solution into one of immense complexity
[1] https://marcinciura.wordpress.com/2023/10/14/orinoko/
Maybe this was not your point, but the difference with the hat/spectre tile is that there is no periodic tiling (with a reflected hat).
Last year, I spend some time generating irregular/random tiling made out of squares and triangles. One of the blogs about it is: https://www.iwriteiam.nl/D2208.html#23b
In 2016, I investigate Versaille like tilings: https://www.iwriteiam.nl/D1606.html#13
In 2003, I spend time to investigate regulat rule-30 tilings. See visualization from 2019: https://www.iwriteiam.nl/D1910.html#23
I was about to ask if the code was available, but then found the programs page. Thanks.
But a correction: the basic shape of the spectre is 14 sides with 2 of them colinear. It's only truly aperiodic when you modify the 14 edges, making it clear it is 'really' 14 sided not 13.