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In combinatorial mathematics, an Aztec diamond of order ''n'' consists of all squares of a square lattice whose centers (''x'',''y'') satisfy |''x''| + |''y''| ≤ ''n''. Here ''n'' is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both ''x'' and ''y'' are half-integers. The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order ''n'' is 2''n''(''n''+1)/2. The arctic circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle. Diamant azteque.svg|An Aztec diamond of order 4, with 1024 domino tilings Diamant azteque plein.svg|One possible tiling. Arctic Circle.svg|A tiling of a hexagon chosen uniformly at random, with the "frozen" tiles being depicted in white. (Arctic Circle theorem.) It is common to color the tiles in the following fashion. First consider a checkerboard coloring of the diamond. Each tile will cover exactly one black square. Vertical tiles where the top square covers a black square, is colored in one color, and the other vertical tiles in a second. Similarly for horizontal tiles. ==References== 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「Aztec diamond」の詳細全文を読む スポンサード リンク
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