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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''.〔B. Grünbaum and G.C. Shephard, ''Tilings and Patterns'', Freemann, NY 1986〕 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. *Any finite region (patch) 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. *They are quasicrystalline: implemented as a physical structure an Ammann–Beenker tiling will produce Bragg diffraction; the diffractogram reveals both the underlying eightfold symmetry and the long-range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation." Various methods to describe the tilings have been proposed: matching rules, substitutions, cut and project schemes 〔Beenker FPM, Algebraic theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus, TH Report 82-WSK-04 (1982), Technische Hogeschool, Eindhoven〕 and coverings.〔F. Ga¨hler, in Proceedings of the 6th International Conference on Quasicrystals, edited by S. Takeuchi and T. Fujiwara, World Scientific, Singapore, 1998, p. 95.〕〔(S. Ben Abraham and F. Gahler, Phys. Rev. B60(1999)860 )〕 In 1987 Wang, Chen and Kuo announced the discovery of a quasicrystal with octagonal symmetry.〔Wang N., Chen H. and Kuo K., Phys Rev Lett. 59(1987) 1010〕 ==Description of the tiles== The most common choice of tileset to produce the Ammann–Beenker tilings includes a rhombus with 45- and 135-degree angles (these rhombi are shown in blue in the diagram at the top of the page) and a square (shown in white in the diagram above). The square may alternatively be divided into a pair of isosceles right triangles. (This is also done in the above diagram.) The matching rules or substitution relations for the square/triangle do not respect all of its symmetries, however. In fact, the matching rules for the tiles do not even respect the reflectional symmetries preserved by the substitution rules. This is the substitution rule for the usual tileset. 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.〔Bédaride N., Fernique Th., ''The Ammann-Beenker Tilings Revisited'' (arXiv )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ammann–Beenker tiling」の詳細全文を読む スポンサード リンク
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