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An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011.〔(【引用サイトリンク】title=The Nobel Prize in Chemistry 2011 )〕 However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. The most frequent ones are explained below. ==Definition and illustration== Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order to rule out such boring examples, one defines an aperiodic tiling to be one that does not contain arbitrary large periodic parts. A tiling is called aperiodic if its hull contains only non-periodic tilings. The hull of a tiling contains all translates ''T+x'' of ''T'', together with all tilings that can be approximated by translates of ''T''. Formally this is the closure of the set in the local topology. In the local topology (resp. the corresponding metric) two tiles are -close if they agree in a ball of radius around the origin (possibly after shifting one of the tilings by an amount less than ). To give an even simpler example than above, consider a one-dimensional tiling ''T'' of the line that looks like ...''aaaaaabaaaaa''... where ''a'' represents an interval of length one, ''b'' represents an interval of length two. Thus the tiling ''T'' consists of infinitely many copies of ''a'' and one copy of ''b'' (with centre 0, say). Now all translates of ''T'' are the tilings with one ''b'' somewhere and ''a''s else. The sequence of tilings where ''b'' is centred at converges - in the local topology - to the periodic tiling consisting of ''a''s only. Thus ''T'' is not an aperiodic tiling, since its hull contains the periodic tiling ...''aaaaaa''.... For many well-behaved tilings (e.g. substitution tilings with finitely many local patterns) holds: if a tiling is non-periodic and repetitive (i.e. the each patch occurs uniformly dense throughout the tiling) then it is aperiodic.〔 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「Aperiodic tiling」の詳細全文を読む スポンサード リンク
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