Wow!! What a breathe of fresh air this paper is in the midst of suffocating levels of "AI solves everything" hype cycle.

https://arxiv.org/abs/2303.10798

They have found at long last, a single tile, an "einstein", which they call a "hat"/polykite that tiles the entire plane aperiodically.

Previously the best known aperiodic tiling of the plane required at the least two different tiles, the most famous ones being the Penrose tiles, and those that adorn Alhambra.

It is all the more wonderful that the first two authors don't have any academic/research affiliations. They write somewhere in the paper, how it all started, so wonderful:

"One of the authors (Smith) began investigating the hat polykite as part of his open-ended visual exploration of shapes and their tiling properties. Working largely by hand, with the
assistance of Scherphuis’s PolyForm Puzzle Solver software (www.jaapsch.net/puzzles/
polysolver.htm), he could find no obvious barriers to the construction of large patches, and yet no clear cluster of tiles that filled the plane periodically."

Why is the study of tilings such a big deal? Well, it hints at and tries to formalize various physics concepts that are of immense interest to many of us (and dare I say, even neuroscientists): quasi crystals!, possible new states of matter, emergent structures from simple units, how symmetries and asymmetries arise, stability of heterogenous media, soft matter physics, order without periodicity, criticality etc., etc.,

On quasi-crystals and their search, applications, uses etc., I recommend the wonderful Paul Steinhardt's book: "The Second Kind of Impossible: The Extraordinary Quest for a New Form of Matter"

#Physics #Maths #Combinatorics #AperiodicTiling #PenroseTiles #Einstein #Emergence #condensedmatter

At least I now understand Penrose tilings. :) #math #geometry #PenroseTiles https://www.quantamagazine.org/nasty-geometry-breaks-decades-old-tiling-conjecture-20221215/
"‘Nasty’ Geometry Breaks Decades-Old Tiling Conjecture"
"Mathematicians predicted that if they imposed enough restrictions on how a shape might tile space, they could force a periodic pattern to emerge. But they were wrong."

I just finished fitting my double set of Perplexing Poultry together. #penrosetiles. I love how the tiles look, they remind me of my hacker pal Bill Gosper and of the hero of Robert Lawson's supernal MCWHIWINNEY'S JAUNT. https://gutenberg.ca/ebooks/lawsonr-mcwhinneysjaunt/lawsonr-mcwhinneysjaunt-00-h-dir/lawsonr-mcwhinneysjaunt-00-h.html

@iain_bancarz I'dlove to see your paper. Trying to use up all my Penrose tiles, I get stuck and then I have to back up and redo part of it. Not clear on how radical a "back up" might potentially be required. 10 steps, 50, arbitrarily large? Consider problem of predicting whether a given Penrose tile configuration can be continued indefiinitely. Might this be computationally unsolvable in same sense as Turing's halting problem? #turingmachine #haltingproblem #penrosetiles #perplexingpoultry

#Penrosetiles #perplexingpoultry #freeware #wares. On a binge with a double set of these funny looking tiles. Non-periodic tiles ... tricky to make them all go in. Invented by the great mathematician Roger Penrose. I have 3D Perplexing Poultry in Chapter Three of my epic cyberpunk masterpiece FREEWARE. https://www.rudyrucker.com/wares/cc_downloads/html/#freeware

Pattern Collider: Neat interactive tool for generating quasiperiodic tilings
https://aatishb.com/patterncollider/
#via:metafilter #penrosetiles #tesselation #geometry #tiles #art #+