3 ms·

Just the one? I have no idea how this is described mathematically, but just looking at the image in the article, the shape spans three hexagons, comprising 2/6

by OJFord 4y ago

Just the one? I have no idea how this is described mathematically, but just looking at the image in the article, the shape spans three hexagons, comprising 2/6 sectors of two of them and 4/6 of the third. I have no idea what I'm talking about, but it seems like it 'ought' to scale to larger (or at least some larger) polygons, or number of them spanned, even excluding trivial multiples (or 6/6 covered ones inserted in the middle) that are effectively the same shape.

Someone 4y ago
There probably are countless others, but this is the first that we know of. And I wouldn’t know whether trivial multiples that still tile the plane non-periodically exist. Once you pick a multiple, even the claim that any of these basic structures in the plane is part of the multiple you picked doesn’t seem to have an obvious, trivial (1) proof to me, let alone the additional requirement that you can find non-overlapping ones. (1) I’m trying, likely unsuccessfully, to dodge the problem of triviality in mathematics (https://en.wikipedia.org/wiki/Triviality_(mathematics) https://en.wikipedia.org/wiki/Triviality_(mathematics)) here
- odgaus 4y ago
  On their project page [1] they even mention that there is a (infinite) family of shapes >> The hat is one member of a continuous family of shapes that are all aperiodic, and that all tile the plane in the same way. [1] https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/
penteract 4y ago
In the paper(linked in other comments), they say it's part of a family of such tiles which can be generated by changing some of the edge lengths.
dclowd9901 4y ago
From a maths standpoint, I’m curious what the analogy to numbers is. Would this tile be like an irrational number? A prime number? But in 2d space?
- hgsgm 4y ago
  In 1D you can't have a single shape tile aperiodically, since there is no room for variation. You could make a 2D diagram of the integers with they prime factorizations, which is aperiodic, but nearly periodic, but requires an infinite set of different "tiles". Perhaps you could take an irrational or transcendebtal number, and take its multiples or powers mod 1, to get an aperiodic nearly periodic sequence.

by OJFord 4y ago

Someone 4y ago

There probably are countless others, but this is the first that we know of. And I wouldn’t know whether trivial multiples that still tile the plane non-periodically exist. Once you pick a multiple, even the claim that any of these basic structures in the plane is part of the multiple you picked doesn’t seem to have an obvious, trivial (1) proof to me, let alone the additional requirement that you can find non-overlapping ones. (1) I’m trying, likely unsuccessfully, to dodge the problem of triviality in mathematics (https://en.wikipedia.org/wiki/Triviality_(mathematics) https://en.wikipedia.org/wiki/Triviality_(mathematics)) here

odgaus 4y ago
On their project page [1] they even mention that there is a (infinite) family of shapes >> The hat is one member of a continuous family of shapes that are all aperiodic, and that all tile the plane in the same way. [1] https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/

penteract 4y ago

In the paper(linked in other comments), they say it's part of a family of such tiles which can be generated by changing some of the edge lengths.

dclowd9901 4y ago

From a maths standpoint, I’m curious what the analogy to numbers is. Would this tile be like an irrational number? A prime number? But in 2d space?

hgsgm 4y ago
In 1D you can't have a single shape tile aperiodically, since there is no room for variation. You could make a 2D diagram of the integers with they prime factorizations, which is aperiodic, but nearly periodic, but requires an infinite set of different "tiles". Perhaps you could take an irrational or transcendebtal number, and take its multiples or powers mod 1, to get an aperiodic nearly periodic sequence.