Sometimes call the "hat" or the "t-shirt" when viewed upside down. Also called "einstein", a pun on the German phrase “ein stein,” which means “one piece.” post 
Printable design. thingiverse
pages/aperiodic-monotile
My STL viewer doesn't see this part. Maybe Thingiverse sent me a binary version of STL? Maybe I need a new viewer? github site
http://scad.fed.wiki/assets/viewstl/viewstl.html?stl=http://scad.fed.wiki/assets/pages/aperiodic-monotile/AperiodicTile.stl HEIGHT 370
My printer could read the file and printed a nicely sized part for handling. I hope to thus develop an intuition that informs reading of the proofs.
Casual assembly leads to voids not easily filled.
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Mathematically, this leaves open the question of whether a single shape can force aperiodicity using only translations and rotations. Spectres is closely related to a tiling with a sparse distribution of hats lying within a dense field of turtles, and one with a sparse distribution of turtles lying within a dense field of hats. post mastodon
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An 8-D Lattice cut by a skew plane lying through it a 2-face gets projected onto the plane iff its dual 6-face intersects the plane. To check this I take all the 5-faces bounding the 6-face and check their signed distance to the plane. mastodon
A fundamental result in the theory of nonperiodic tilings was the discovery of the fact that some substitution tilings can be obtained by projecting certain points from higher dimensional point lattices. post
In this paper we put the work of Professor N.G. de Bruijn on quasicrystals in historical context. pdf