{"id":291,"date":"2015-10-26T10:10:23","date_gmt":"2015-10-26T10:10:23","guid":{"rendered":"http:\/\/www.hibma.org\/wpaperiodictiling\/?page_id=291"},"modified":"2017-01-07T09:53:07","modified_gmt":"2017-01-07T09:53:07","slug":"harriss","status":"publish","type":"page","link":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/harriss\/","title":{"rendered":"Harriss Tiles"},"content":{"rendered":"

\n

A generalization of a rhomb tiling<\/a> for arbitrary \"n>3\" was introduced by E.O. Harriss. It\u00a0is based on\u00a0an edge sequence of \"(0 ,1 ,-1 ,0)\".\u00a0Using our treatment and notation \u00a0the basic substitution tile matrix for this edge sequence is<\/p>\n

\n

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

To reduce the size of the\u00a0prototile set for odd n, Harriss used a substitution rule in which the \"s\pm1\" tiles are replaced by the \"n-s\mp1\" tiles or<\/p>\n

\n

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

Note that the pair of \"s\"-tiles in the center have to be\u00a0rotated over \"\pi\" in order to fit in for higher generations.\u00a0The twofold rotation is signified by underlining the matrix entry.<\/p>\n

In the figure below the substitution tiles for \"n=5\" are shown for both cases.<\/p>\n

\"\"<\/a>
Fig. H1. Comparison of our basic substitution rule for n=5 and an edge sequence \"(0 ,1 , -1, 0)\" and Harriss’s substitution rule for odd \"n\" and even \"s\".<\/figcaption><\/figure>\n

To apply the basic substitution rule eq. (1)\u00a0all possible prototiles, the positive as well as\u00a0the negative ones, have to be included. The tiles in the \"s=4\" tile can be rearranged to avoid the negative \"s=6\" tile, but it is not possible to circumvent the \"s=7\" negative tile in the \"s=5\" substitution tile. And although not visible the \"s=5\" tile (a straight line) has to be used in the \"s=4\" substitution tile as will become apparent in the next generation supertiles.<\/p>\n

For odd \"n\", Harriss’s substitution rule separates the prototiles into two groups, the odd and even s prototiles. Formally,\u00a0the zero area and negative tiles are also involved. Nevertheless, the set of even tiles may be reduced to two positive tiles, the \"s=2\" and the \"s=4\" tiles, by neglecting the \"s=0\" tile, and by \u00a0letting the \"s=6\" negative tile annihilate a \"s=4\" tile and a rearrangement of the remaining tiles. In the set of odd s tiles, \"s=1 , 3, 5\", the\u00a0\"s=5\" tile\u00a0has\u00a0to be used.\u00a0Because\u00a0this tile sonsists of congruent negative and positive parts, its role is to move a patch of tiles\u00a0from a n overlapping to an empty part of the substitution tile. Harriss’s solution was to add substitution tiles with additional\u00a0or\u00a0missing prototiles. Our treatment gives the same result, but in a simpler, more straightforward way.<\/p>\n

There are many other Harriss-like tilings without negative tiles for edge sequences with one or more \"\pm1\" pairs and starting with a 0 to avoid crossing of neighboring edges.<\/p>\n

(0 ,1,-1) edge with substitution tile matrix:<\/p>\n

\n

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

The right column and last row \u00a0of the Harriss tile have been removed. The inflation factor is \"1+2\cos{\pi/n}\"<\/p>\n

(0 ,1 ,0 ,-1 ,0) edge with substitution tile matrix:<\/p>\n

\n

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

An extra row and column of prototiles have been added in\u00a0the middle of the \u00a0Harriss tile. In this case\u00a0a rearrangement of the prototiles \"s= 4\" is not even needed.\u00a0The inflation factor is \"3+2\cos{\pi/n}\".<\/p>\n

A second set of substitution tiles with a different edge shape (or substitution tile edge sequence) may be obtained by rotating all prototilesover \"\pi\" in the above Harriss-like substitution tiles.<\/p>\n

\"Harriss-like<\/a>
Fig. H2. Harriss-like tilings<\/figcaption><\/figure>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

A generalization of a rhomb tiling for arbitrary was introduced by E.O. Harriss. It\u00a0is based on\u00a0an edge sequence of .\u00a0Using our treatment and notation \u00a0the basic substitution tile matrix for this edge sequence is (1)   To reduce the size of the\u00a0prototile set for odd n, Harriss used a substitution rule in which the tiles … Continue reading Harriss Tiles<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":6,"comment_status":"closed","ping_status":"closed","template":"page-templates\/full-width.php","meta":{"footnotes":""},"class_list":["post-291","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages\/291","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/comments?post=291"}],"version-history":[{"count":35,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages\/291\/revisions"}],"predecessor-version":[{"id":571,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages\/291\/revisions\/571"}],"wp:attachment":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/media?parent=291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}