page_title: Ammann–Beenker tiling
section_title: Description of the tiles
hierarchical_section_title: Ammann–Beenker tiling / Description of the tiles
caption_attribution_description: English: Amman's A and B pair of A5 tiles, decorated with matching rules; any tiling by these tilings is necessarily non-periodic, and the tiles are therefore aperiodic.
context_page_description: In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker.
  Because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings discovered by Ammann and described in Tilings and Patterns.
  The Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings, most notably:
  They are nonperiodic, which means that they lack any translational symmetry.
  Their non-periodicity is implied by their hierarchical structure: the tilings are substitution tilings arising from substitution rules for growing larger and larger patches. This substitution structure also implies that:
  Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, if one looks only at finite patches.
context_section_description: Amman's A and B tiles in his pair A5 a 45-135-degree rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic, hierarchical, and quasiperiodic structures of each of the infinite number of individual Ammann–Beenker tilings. 
  An alternate set of tiles, also discovered by Ammann, and labelled "Ammann 4" in Grünbaum and Shephard, consists of two nonconvex right-angle-edged pieces. One consists of two squares overlapping on a smaller square, while the other consists of a large square attached to a smaller square. The diagrams below show the pieces and a portion of the tilings.
   This is the substitution rule for the alternate tileset.
   The relationship between the two tilesets.
  In addition to the edge arrows in the usual tileset, the matching rules for both tilesets can be expressed by drawing pieces of large arrows at the vertices, and requiring them to piece together into full arrows.
  Katz has studied the additional tilings allowed by dropping the vertex constraints and imposing only the requirement that the edge arrows match. Since this requirement is itself preserved by the substitution rules, any new tiling has an infinite sequence of "enlarged" copies obtained by successive applications of the substitution rule. Each tiling in the sequence is indistinguishable from a true Ammann–Beenker tiling on a successively larger scale. Since some of these tilings are periodic, it follows that no decoration of the tiles which does force aperiodicity can be determined by looking at any finite patch of the tiling. The orientation of the vertex arrows which force aperiodicity, then, can only be deduced from the entire infinite tiling.
  The tiling has also an extremal property : among the tilings whose rhombuses alternate (that is, whenever two rhombuses are adjacent or separated by a row of square, they appear in different orientations), the proportion of squares is found to be minimal in the Ammann–Beenker tilings.
caption: Amman's A and B pair of A5 tiles, decorated with matching rules; any tiling by these tilings is necessarily non-periodic, and the tiles are therefore aperiodic.
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