A Brief History of Tricky Mathematical Tiling | Hacker News Reader

A Brief History of Tricky Mathematical Tiling

quantamagazine.org

45 pointsColinWright14 comments

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

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

Very 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.

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.

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].

[1] https://marcinciura.wordpress.com/2023/10/14/orinoko/

Nice 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).

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 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?

https://en.wikipedia.org/wiki/Frequency_illusion perhaps?

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 spectre has 13 sides. Is it known whether this is the minimum possible for an einstein that only tiles aperiodically and without reflection?

I'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.

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.

lozenge 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

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