I suppose that most of you already heard about the first ever aperiodic planar tiling with one type of tiles. It was discovered by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. Amazing!!!

Here are  blogposts from Complex Projective 4-Space and from the Aperiodical.

An aperiodic monotile

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

Abstract:  A longstanding open problem asks for an aperiodic monotile, also known as an “einstein”: a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the “hat” polykite, can form clusters called “metatiles”, for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical — and hence aperiodic — tilings.

Comments:  

The new paper starts with a description of the very interesting history. From Hilbert’s Entscheidungsproblem, that led to notions of undecidability, Wang’s 1961 work, and Berger’s 1966 undecidability result that showed that some tiles admit only apriodic tilings. (This required 20426 types of tiles.) Penrose’s famous apreiodic tiling from 1978 consists of just two tiles.

There are lovely related results for the hyperbolic plane starting from  1974 paper by Boroczky, and later papers by  Block and Weinberger, Margulis and Mozes, Goodman-Strauss and others.

I could not figure out from browsing the paper about the status of the question: “Can we have aperiodic tiling with convex tiles?” 

Half a year ago Rachel Greenfeld and Terry Tao disproved in the paper  “A counterexample to the periodic tiling conjecture,“ the periodic tiling conjecture (Grünbaum-Shephard and Lagarias-Wang) that asserted that any finite subset of a lattice which tiles that lattice by translations, in fact tiles periodically.