Aperodic Mysteries

If I give you a collection of shapes that fit together, it is natural to ask if they can keep tiling, or is there a limit to how far they can go? For some sets of shapes they can go on forever, but no matter how you fit them together the resulting tilings will never repeat exactly. These shapes are called aperiodic tile sets, and they produce some beautiful tilings, the Penrose tiling is a great example, but more recently a team, including Chaim Goodman Strauss, who was at the University of Arkansas, found examples with just a single shape (the hat and spectre tiles) answering an open mathematical problem that was over 50 years old!

Videos:

• What are aperiodic tilings and the discovery of an aperiodic monotile:
  • https://www.youtube.com/watch?v=IfVwelta1fE

• Discussion of the process of discovery with Craig Kaplan, one of the team behind the discovery:
  • https://www.youtube.com/watch?v=_ZS3Oqg1AX0

• A discussion  of aperiodic tiling and the Penrose tiling, from before the discovery of a monotile:
  • https://www.youtube.com/watch?v=48sCx-wBs34

Additional Resources:

• The original paper revealing the Penrose tiling:
  • Penrose Tiles to Trapdoor Ciphers by Martin Gardner
    https://maa.org/sites/default/files/pdf/pubs/focus/Gardner_PenroseTilings1-1977.pdf

• A discussion of the history of tiling problems:
  • A Brief History of Tricky Mathematical Tiling by Dave Richeson
    https://www.quantamagazine.org/a-brief-history-of-tricky-mathematical-tiling-20231030/

• The role of a community in finding the hat and spectre tiles by one of the team behind the discovery:
  • The Hat tile and Community in Mathematics by Chaim Goodman Strauss and Casey Mann
    https://digitaleditions.walsworth.com/publication/?m=7656&i=799289&p=30&ver=html5

Technical Resources:

• An overview of the mathematics of pattern and order, the “aperiodic” in the title is different to that discussed here, but it includes the study of the tilings for many aperiodic sets of tiles:
  • Aperiodic Order, by Michael Baake and Uwe Grimm
    http://www.aperiodicorder.org/

• The original paper with the Spectre monotile:
  • A chiral aperiodic monotile by David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss:
    https://arxiv.org/abs/2305.17743