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 dendrite 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 dendrite rule initiates a new method to produce tilings of the plane.

Forcing nonperiodic tilings with one tile using a seed

“…Such a combination of a seed and a classical matching rule is not a new concept in the context of nonperiodic tilings (e.g. in [7] or [8]).…”

“…So, for most of the sectors the number of tiles per row is denoted by (1, 3, 5, ...) and three of them -at the left half -are characterized by (3,5,7, ...), i.e., the leading single tile was skipped, but apart from this first tile all sectors are congruent. We can observe that each boundary between any two sectors is an unlimited curve periodically meeting a straight line (see the red lines in Fig.…”

“…However, a certain order has to be kept in form of a sequence of local checks. A similar rule where small subtrees ("dendrites") are added to a connected tree is used in [7].…”

“…Rule type 1: Matching rules for adjacent tiles (no colors required) [9] Rule type 2: Colors required for matching adjacent tiles [10] Rule type 3: Non-adjacent but pairwise matching [11] Rule type 4: Configurations of adjacent tiles from a catalogue [2] Rule type 5: Adjacent, following a sequence or tree ( [7] or above cases) Rule type 6: A given number of translation classes must be found. Rule type 7: A patch of tiles must exist in the tiling ( [8] and see below).…”

“…In 2011 studies on the "einstein problem" (in German "ein Stein" = "one stone") came to a remarkable result, when Socolar and Taylor [11] presented their decorated single prototile forcing nonperiodicity of any generated tiling by two matching rules. Other attempts were made also with one tile and two matching rules [7]. The ideal "einstein" would be a single tile (a topological disk) only allowing nonperiodic tilings without need of any matching rule.…”