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A NOTE ON APERIODIC AMMANN TILES SHIGEKI AKIYAMA
 

Summary: A NOTE ON APERIODIC AMMANN TILES
SHIGEKI AKIYAMA
Abstract. We present a variant of Ammann tiles consisting of two
similar rectilinear hexagons with edge subdivision, which can tile the
plane but only in non periodic ways. A special matching rule, ghost
marking, plays a key role in the proof.
In this note, a tile is a closed polygon in R2. Denote by A a finite set
of tiles. A tiling T of R2 is a collection of tiles which covers R2 without
interior intersections, and each tile is congruent to an element of A under
rigid motion (action of isometry of R2: composition of rotation, translation,
and reflection). A tiling T has a period p R2 when the tiling exactly
matches with its translation by p. A patch of a tiling T is a finite set of
tiles in T whose union is homeomorphic to a closed ball. A tiling T is non
periodic if the only period of T is 0. A set of tiles A is called aperiodic if A
generates a tiling, but all the tilings generated by A are non periodic. To
enforce aperiodicity, we often introduce certain matching rules (conditions)
on the surfaces of tiles. A well known example of aperiodic tiles is due to
Penrose which consists of two kinds of tiles: kites and darts with matching
rules. Ammann gave several sets of aperiodic tiles, one of which is a target
of this paper. See [9, 14] for tilings by Penrose, Ammann and more with

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics