An aperiodic monotile that forces nonperiodicity through dendrites

Abstract

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.

An aperiodic tile with edge-to-edge orientational matching rules

“…Indeed, we may choose an arbitrary charge for one edge and then assign charges to others by moving through G applying the above two rules; we do not encounter inconsistencies because each path-component of G is a tree. In fact, each path component has exactly the tree structure defined by the growth condition in [8]. We claim that a tiling satisfying R1 and the above two conditions also satisfies R2.…”

“…Belonging to the hull is a local (in fact, edge-to-edge) condition and as a consequence the hull is a compact space. This should be contrasted to the tilings of [8], where the valid tilings do not form a compact space. In [8], valid tilings are those which can be constructed from a growth rule which is local, but whether a given complete tiling is in the hull is a non-local condition that depends on whether or not an embedded tree is connected.…”

“…This should be contrasted to the tilings of [8], where the valid tilings do not form a compact space. In [8], valid tilings are those which can be constructed from a growth rule which is local, but whether a given complete tiling is in the hull is a non-local condition that depends on whether or not an embedded tree is connected. We note that every tiling permitted in [8] is also a valid tiling for our current tile by adding charge conditions and reflections as appropriate.…”