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Comment by shireboy | Hacker News Reader

I'm confused - I see multiple repeating patterns. three light blue hats in triangle around dark blue. grey boomerang pattern. two white tiles with same rotation and layout. I'm sure I misunderstand what is meant by "pattern that never repeats", but please dumb this down for me.

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Those configurations appear often, but they don't appear with a period. Look at a chessboard.. i can make it out of individual black and white tiles, or out of pre-assembled units of 1 white and one black tile (or a strip of length 4 or squares of size 4, etc). I can make a chess board by grouping the base units into a pattern, then only using that pattern. I can put that pattern on a wheel and roll it along a surface forever to get an infinite chessboard pattern.

There's not a way to do that sort of grouping for these tiles.

Compare pi. I can find the numbers representing my name in ascii an infinite number of times in the digits of pi, but if i find it once, there's no information in where to find it again, i can't just move forward n digits to find it, then another n digits to find it again, and ao on.

Thank you for explaining "aperiodic" in a way that makes sense to non-geometers!

"An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches" [1]

So, it has "islands" of repenting combinations of tiles, but these islands do not repeat / translate in a regular way.

1: https://www.wikiwand.com/en/Aperiodic_tiling

If you were to make two copies of the pattern and superimpose them on each other, they would never match perfectly no matter how you rotated them and shifted them around.

However, they might match in small patches, just never across the entire infinite plane.

wait, you must mean if you created a copy on top of another copy, they would match like that, but there's no combination of shifting, rotation, mirroring, etc that would also match? (unless it you shifted it back to the starting orientation)

cuz it would be mind blowing if you made a copy and it wasn't a copy... pauli whackamole exclusion tiling

For example, you can't replace these tiles with just rectangles. For regular shapes you can, for example if you tile your bathroom with squares there will be a repeating 2x2 square pattern too. Similarly, hexagons and triangles can be replaced with rectangular tiles (where every tile is the same). You can't do this for this pattern.

I was about to ask the same question but it appears the repeats are not exactly identical at the edges.

Still, can one prove this is aperiodic from geometry alone? It seems rather difficult to actually prove that fact. Feels intuitive that there must be a period somewhere, however large it may be, on the infinite 2D plane.

For most sets of shapes a periodic tiling is possible, but by no means guaranteed.

For example, consider rectangles with sides of 1 and 3 units: they can cover the plane periodically (e.g. in a simple rectangular grid), but also aperiodically, because you can form a square grid of square 3 by 3 units "metatiles", each encoding one bit of information in the vertical or horizontal orientation of the narrow rectangles; then it's easy to break symmetry by orienting metatiles so that for all integers m and n some metatile differs from the metatile m rows and n columns away, so the period cannot be m rows and n columns.

I’m assuming the infinite sequence of segments along a straight vertical line at each “x-coordinate” (not sure how to say this, but you can see vertical lines made of blocks, mean those) were proved to be unique and non repeating

It looks to me like it repeats in 1 direction pretty quickly, but in other directions doesn't repeat at all... At least, as far as I can tell from the samples given.

The claim is that this tesselation is aperiodic, meaning that the entire pattern does not have translational symmetry.

It only repeats for a little bit.

Whatever the size of a pattern you find, you cannot tile the plane with translations (just translations) using that pattern: you need to rotate it. The definition is just that.

So, there are "patterns" (as a matter of fact, the elementary tile is repeated infinitely on the tiling) but you cannot fill the plane with mere translations of one.

See: https://personal.math.ubc.ca/~cass/courses/m308-02b/projects...

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