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Einstein problem solved (aperiodic monotile discovery)

Ed Pegg, Wolfram Research
Posted 2 years ago

enter image description here

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POSTED BY: Ed Pegg
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Many years ago I (and Tom Sibley) proved that any tiling (finite or infinite) with Penrose rhombs is 3-colorable as a map (a question raised by John Conway). And later the same was proved for Kites and Darts. So I wonder if the same is true for hats. Ed: If you send me a pile of hats I can run my 3-coloring program on it. I even made a carpet showing 3-colored rhombs.
POSTED BY: Stan Wagon

But wait! There's more!

At Hat tilings via HTPF equivalence, we've got a lot more code and patterns. For example, each hat tile cuts up a hexagon in the hexagonal grid into 2 or 3 pieces. The division orientation is either Y or upside-down Y. If you color the Y divisions red, you get the pattern below. Code for that and many other items at the link.

enter image description here
POSTED BY: Ed Pegg

I didn't get a good svg from the app website, I think: ResourceFunction::usermessage : SVGImport::format: "/Users/pbarendse/Downloads/output(2).svg" does not appear to be a valid SVG file. >>
POSTED BY: Peter Barendse

Can the single tile be generated by some kind of morphing of Penrose's kite and dart?
POSTED BY: Paul Abbott
Brad Klee
Brad Klee
Posted 2 years ago
POSTED BY: Brad Klee

It seems from the title image here: https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile that the E. tile can be broken into diamond shapes, each of which could be a kite+dart. So maybe the tile could be broken into kites and darts in a way so that the aperiodicity of the E. tiling would produce aperiodicity in the kites and darts
POSTED BY: Peter Barendse
Brad Klee
Brad Klee
Posted 2 years ago
POSTED BY: Brad Klee

enter image description here -- you have earned Featured Contributor Badge enter image description here Your exceptional post has been selected for our editorial column Staff Picks http://wolfr.am/StaffPicks and Your Profile is now distinguished by a Featured Contributor Badge and is displayed on the Featured Contributor Board. Thank you!
POSTED BY: EDITORIAL BOARD
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