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arXiv:2303.10798math.CO

An aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

Abstract

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

arXiv comments
91 pages, 57 figures. Copyedited journal version of article. Significant editing of the exposition throughout, but particularly in Section 3. Overall the same results appear in the same order
Journal reference
DOI
doi:10.5070/C64163843
Subjects
math.COCombinatorics cs.DMDiscrete Mathematics math.MGMetric Geometry

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Linked by 7 identifiable commenters in 7 comments across 7 threads, from Mar 2023 to Mar 2024. Found in 1 HN submission from Mar 2023 to Mar 2023. Open submission discussions.

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Discussed in comments7 threads

  1. How to Start Google 1 comment

    • We've arrived at a point in the conversation where your assertions are (for the most part) correct in at least a narrow way. But there were several epistemological sleights of hand involved in getting from point A to point B that you as an eminent philosopher can trivially…

      benreesmanView on HNParent context

  2. [dead] 1 comment

    • - Paper site: https://cs.uwaterloo.ca/~csk/hat/ - Paper: https://arxiv.org/abs/2303.10798 - news: https://phys.org/news/2023-06-uk-hobbyist-stuns-math-world.h..., https://www.discovermagazine.com/the-sciences/at-long-last-h...

      codeAlignedView on HNParent context

  3. What can you do with an Einstein (aperiodic monotile)? 1 comment

    • No, I mean mathematically. I had a skim through the original paper[1], but unfortunately it's not easy to figure out what the coordinates are. If any of them are related as a factor of an irrational number, I don't think it's possible. [1] https://arxiv.org/abs/2303.10798

      l0b0View on HNParent context

  4. A Brief History of Tricky Mathematical Tiling 1 comment

    • 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…

      bazzarghView on HNParent context

  5. Hexagonal Grids (2013) 1 comment
  6. Elusive ‘Einstein’ Solves a Longstanding Math Problem 1 comment
  7. Mathematicians discover shape that can tile a wall and never repeat 1 comment

    • Project website with web demo https://cs.uwaterloo.ca/~csk/hat/ Direct link to PDF on ArXiv (89 pages) https://arxiv.org/pdf/2303.10798.pdf

      johndoughView on HNParent context

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