2019/03/04 by Mampusti, Michael, Whittaker, Michael F.
#05B45 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 52C23 #Secondary: 37E25
paper · doi:10.48550/arxiv.1903.01158
We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar--Taylor monotile, but can be realised by shape alone. The second is a local growth rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our local growth rule initiates a new method to produce tilings of the plane.