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A Brief History of Tricky Mathematical Tiling
- move-on-by 3y agoAnyone 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.
- rssoconnor 3y agoI 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.
- ryanschneider 3y agoI just took a break from reading a scene from Anathem revolving around tiling and came across this thread. :) This sort of “HN coincidence” has happened to me several times this week, is there a term for it I don’t know?
- mauricioc 3y agohttps://en.wikipedia.org/wiki/Frequency_illusion https://en.wikipedia.org/wiki/Frequency_illusion perhaps?
- paulpauper 3y agolozenge tiling ...some very complicated formulas for something that seems so simple putting a hole in the center of a tiling grid turns a trivial solution into one of immense complexity
- mci 3y agoI 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]. [1] https://marcinciura.wordpress.com/2023/10/14/orinoko/ https://marcinciura.wordpress.com/2023/10/14/orinoko/
- gilleain 3y agoNice looking shape. 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).
- fjfaase 3y agoI 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 https://www.iwriteiam.nl/D2208.html#23b In 2016, I investigate Versaille like tilings: https://www.iwriteiam.nl/D1606.html#13 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 https://www.iwriteiam.nl/D1910.html#23
- gilleain 3y agoVery nice - I like the irregular square/triangles and the Versailles-like ones. I was about to ask if the code was available, but then found the programs page. Thanks.
- adkaplan 3y agoInteresting! 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.
- gilleain 3y agoApparently M. C. Escher liked the snub square tiling. Well it says that on https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling 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".
- tromp 3y agoThe spectre has 13 sides. Is it known whether this is the minimum possible for an einstein that only tiles aperiodically and without reflection?
- robinhouston 3y agoI'm sure it is not known. We know almost nothing about this class of tiles, except for the very recent discovery that the class is non-empty.
- bazzargh 3y agoThe original monotile paper https://arxiv.org/abs/2303.10798 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.