| | | Professor Rachel Greenfeld
~ Tiling by translations ~
Multiple congruent copies of a single Hat tile cover the Euclidean plane aperiodically by using translations, rotations and reflections of that single tile. However, following the announcement of the discovery of the Hat, some mathematicians expressed the view that allowing reflections meant it was not a “true” monotile. Amazingly, David Smith found a second tile, the Spectre, that did not need the reflection property. Against this backdrop, Rachel Greenfeld's talk began by also removing rotations and asking what could be said when only translations are permitted. She had in mind “The Periodic Tiling Conjecture”. Roughly this claims that if an n dimensional monotile tiles an n dimensional space by translations alone then that tiling is periodic. For a square and a cube this seems obviously true. For higher dimensional hypercubes “Keller's Conjecture” (1930) claimed it remained true. However, Rachel informed us that in 1992 Lagarias and Shor showed it was false in ten or more dimensions, subsequently reduced to eight. Working with Terence Tao, she has generalised this result further to show that for sufficiently high dimension the Periodic Tiling Conjecture is also false. This is via a counterexample based upon Sudoku! Her paper with Terence Tao is on arXiv.
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