{"id":12,"date":"2015-10-12T10:31:21","date_gmt":"2015-10-12T10:31:21","guid":{"rendered":"http:\/\/www.hibma.org\/wpaperiodictiling\/?page_id=12"},"modified":"2020-07-06T13:35:38","modified_gmt":"2020-07-06T13:35:38","slug":"2x2-supertiles","status":"publish","type":"page","link":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/2x2-supertiles\/","title":{"rendered":"Supertiles."},"content":{"rendered":"

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A. 2×2 integer edge sequence solution.<\/h2>\n

The smallest first generation supertile is one based on either the \"(1, -1)\" or the (\"\frac{1}{2}\", \"-\frac{1}{2}\") edge sequence. \u00a0In fact, the latter may be considered to be a special case of the first and will be discussed later.\u00a0The \"(1,-1)\" edge sequence means that there is only one \"\pm\" pair of edge angles, or \"m_1=1\". It follows that \"n_0=2\" and \"n_2=1\", i.e. the supertile \"S_s^n\" has to be constructed from the set of prototiles {\"2T_s^n\",\"T_{s+2}^n\", \"T_{s-2}^n\"}. In our general model the arrangement of the tiles is given by the supertile matrix<\/p>\n

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(1) <\/span>   <\/span>\"\begin{equation*}S_s^n=\begin{bmatrix}T_s^n &T_{s+2}^n \\T_{s-2}^n &T_s^n \\\end{bmatrix}\end{equation*}\"<\/p>\n<\/p>\n

In order to tile the plane, the inflation factor should be larger than one.\u00a0Because \"L = 2\cos(\pi{}/n)\", \"n\" should be larger than three.\u00a0Below, supertiles are shown for \"n=4,5\" and \"6\". Rhomb prototiles \"T_s^n\" and \"T_{n-s}^n\"\u00a0<\/sub>are identical objects, but supertiles \"S_s^n\" and \"S_{n-s}^n\" are not. The supertiles also loose the twofold rotation axis perpendicular to the plane of the prototiles. Consequently, after the first generation supertiles there are four different substitution rules for the next supertile generation, which we have denoted by a), b), c) and d). a) is the basic supertile, in b) the tiles of a) have been rotated over \"\pi{}\",\u00a0in c) the \"s-2\" and \"s+2\" tiles of case a) have been replaced by the \"n-s+2\" and the \"n-s-2\" tiles respectively, and in d) the tiles of case c) have been rotated over \"\pi{}\". In matrix notation:<\/p>\n

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(2) <\/span>   <\/span>\"\begin{equation*}\begin{bmatrix}T_s^n &T_{s+2}^n \\T_{s-2}^n &T_s^n \\\end{bmatrix} , \begin{bmatrix}\underline{T_s^n} &\underline{T_{s+2}^n} \\\underline{T_{s-2}^n }&\underline{T_s^n }\\\end{bmatrix} , \begin{bmatrix}\underline{T_s^n} &T_{n-s-2}^n \\T_{n-s+2}^n &\underline{T_s^n} \\\end{bmatrix} , \begin{bmatrix}T_s^n &\underline{T_{n-s-2}^n} \\\underline{T_{n-s+2}^n} &T_s^n \\\end{bmatrix}\end{equation*}\"<\/p>\n<\/p>\n

Underlined entries denote prototiles\u00a0rotated over \"\pi\".<\/p>\n<\/div>\n<\/div>\n