What Islamic tiling patterns are constructible? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:13:51Z http://mathoverflow.net/feeds/question/35526 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35526/what-islamic-tiling-patterns-are-constructible What Islamic tiling patterns are constructible? Brian Wichmann 2010-08-13T19:27:30Z 2010-08-14T00:38:42Z <p>Eric Broug in his book <em>Islamic Geometric Patterns</em> gives straightedge and compass construction of some simpler patterns. It is clear his techniques will provide constructions for many Islamic patterns.</p> <p>Looking at formal constructibility, the Wikipedia pages gives Gauss' result that 7, 9, 11, 13, 14, 18... etc sided polygons are not constructible. Hence the pattern</p> <p><a href="http://tilingsearch.org/HTML/data160/J43C.html" rel="nofollow">http://tilingsearch.org/HTML/data160/J43C.html</a></p> <p>is not constructible since it contains a regular 9-pointed star polygon.</p> <p>I have over 800 Islamic patterns on my web site but I use a computer and trigonometry to produce my images. It seems that about 40 Islamic patterns on my site are not constructible.</p> <p>Given an Islamic pattern that is not excluded from construction by Gauss' result, it is almost certainly constructible if the following is true:</p> <blockquote> <p>Given two points on the plane, a polygon ($n$ sides) can be constructed with the two points as an edge, provided $n$ is not equal to 7, 9, 11, 13, 14, 18... etc.</p> </blockquote> <p>This result would allow patterns to be built up piece-by-piece.</p> <p>EDIT, Will Jagy: from his profile page, the OP's website, in this address preset to display the tilings in a slideshow format on a web browser, is at</p> <p><a href="http://www.tilingsearch.org/" rel="nofollow">http://www.tilingsearch.org/</a></p> http://mathoverflow.net/questions/35526/what-islamic-tiling-patterns-are-constructible/35532#35532 Answer by Will Jagy for What Islamic tiling patterns are constructible? Will Jagy 2010-08-13T19:40:35Z 2010-08-13T19:40:35Z <p>Yes, that is right. It seems all you are missing is this: given a number of sides $n$ such that the regular polygon of $n$ sides is constructible (by the results of Gauss and Wantzel), how to force the edge length to be a fixed length, call it $L$?  All you need to do is construct the regular $n$-gon. Draw the perpendicular bisector of any edge, which is then a line that passes through the center of the regular polygon, and stop the line at that center so that it is actually a half-line, a "ray." Call this ray $r.$ The ray $r$ starts in the interior of one central triangle of the regular polygon, call the radii acting as the two othe edges ot that triangle $e_1$ and $e_2$  Now, parallel to $r,$ construct the parallel line at distance $$\frac{L}{2}$$ from $r.$ At some point $P,$ the ray $r$ intersects either $e_1$ or $e_2.$ The point $P$ is a vertex of the polygon you want. To find the rest of the polygon just draw a circle with center at the center of the polygon and passing through $P,$ then extend the radii passing through the vertices of your original polygon until they reach the circle. </p>