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A chiral aperiodic monotile

A chiral aperiodic monotile

2023-05-30 01:16:45


A chiral aperiodic monotile


David Smith, Joseph Samuel Myers, Craig S. Kaplan, and
Chaim Goodman-Strauss, 2023



The “hat” aperiodic monotile
resolves the query of whether or not a single form can drive aperiodicity
within the airplane. Nonetheless, all tilings by the hat require reflections; that
is, they have to incorporate
each left- and right-handed hats. Mathematically, this
leaves open the query of whether or not a single form can drive aperiodicity
utilizing solely translations and rotations. (It additionally complicates the
sensible software of the hat in some ornamental contexts, the place additional
work could be wanted to fabricate each a form and its reflection.)

On this paper we current a form that resolves the query above:
one which tiles the airplane aperiodically with out reflections. Particularly,
we present that
the equilateral polygon referred to in our first paper as Tile(1,1)
is a weakly chiral aperiodic monotile: should you merely forbid
reflections by fiat then it admits solely non-periodic tilings (despite the fact that
it tiles periodically should you enable reflections). We then modify
the perimeters of Tile(1,1) to supply a household of shapes we name “Spectres”,
that are strictly chiral aperiodic monotiles: they tile
aperiodically utilizing solely translations and rotations, even when reflections
are permitted.

This web page collects the sources related to this work.

Instruments and hyperlinks

  • A preprint of the paper
    is accessible on the arXiv. (Will probably be posted when out there)

  • You possibly can create your personal patches of Tile(1,1), and save them as PNG
    recordsdata, utilizing an interactive application
    that runs in your internet browser.

  • You possibly can obtain an SVG file
    containing outlines of three variations of the Spectre in order for you
    to chop out copies and experiment with them.
  • Each tiling by Spectres is carefully associated to a tiling with a
    sparse distribution of hats mendacity inside a dense area of turtles,
    and one with a sparse distribution of turtles mendacity inside a dense
    area of hats. You possibly can watch
    a short animation
    that demonstrates this equivalence by morphing
    constantly between these three tilings. You too can
    download your own copy
    (which you’ll extra simply watch looped).

Pattern pictures

Listed below are some pattern pictures you should utilize in publications, media, and so forth.
Be at liberty to change these pictures to fit your tastes.

Creative Commons License
All pictures, and the MP4 animation above, are licensed underneath a Creative Commons Attribution 4.0 International License.

 

 

See Also

A duplicate of Determine 2.2 of the paper, displaying 5 generations of the
Spectre cluster after making use of our substitution guidelines.

[1775×2000 PNG] [Scalable PDF]

 

A looping animated GIF displaying the identical equivalence described above between
tilings by Spectres and tilings by mixtures of hats and turtles.

[500×500 GIF]

 

If you want to contact us about this paper, please e-mail me at
csk@uwaterloo.ca.

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