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A chiral aperiodic monotile

arxiv.org

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Re: A chiral aperiodic monotile

#2

Post by abetusk »

The recently discovered 2D "hat" monotile tiles aperiodically but allows reflections (and must allow reflections to tile?).

The paper discusses a (2D) monotile whose shape will allow an aperiodic tiling without reflections.

Re: A chiral aperiodic monotile

#5

Post by troymc »

onos wrote: post #3

Can anyone explain why people are interested in tilings?

Tilings are easy to understand and relate to... you can tile your bathroom floor with them for example. Despite that superficial banality, it turns out that it takes a lot of mathematical cleverness to construct and analyze them fully. This particular result is one that people have been chasing for decades and has involved some of the smartest mathamaticians in the world, including Conway and Penrose.

Re: A chiral aperiodic monotile

#6

Post by jameshart »

abetusk wrote: post #2

The recently discovered 2D "hat" monotile tiles aperiodically but allows reflections (and must allow reflections to tile?). The paper discusses a (2D) monotile whose shape will allow an aperiodic tiling without reflections.

Notably, it's the same authors: David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

Re: A chiral aperiodic monotile

#7

Post by onos »

troymc wrote: post #5
onos wrote: post #3

Can anyone explain why people are interested in tilings?

Tilings are easy to understand and relate to... you can tile your bathroom floor with them for example. Despite that superficial banality, it turns out that it takes a lot of mathematical cleverness to construct and analyze them fully. This particular result is one that people have been chasing for decades and has involved some of the smartest mathamaticians in the world, including Conway and Penrose.

I see thanks. I guess, hearing about other famous hard problems, I am used to seeing comments like: “if problem X could be solved it would unlock lots of other important areas of math.” Wondering if this tiling area draws attention for its own sake alone or if the problems here are similarly vectors to attack larger issues.

Re: A chiral aperiodic monotile

#8

Post by TrueDuality »

Hmmm... I wonder if these spectres are potentially a basis for a new form of cryptographic algorithms. Unique non-repeating sequences exclusively derived from a set of rules and an initial state in multiple dimensions sounds like a promising candidate.

Re: A chiral aperiodic monotile

#9

Post by firstlink »

I wonder if the authors anticipated this result when publishing their first paper, or if they were primarily motivated by "complaints" that their hat tile (and other tiles in the associated spectrum) required reflection. Certainly they mention this question, but my question is whether they completely anticipated it.
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