A question about tilings of the plane - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:11:12Z http://mathoverflow.net/feeds/question/28556 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28556/a-question-about-tilings-of-the-plane A question about tilings of the plane Garabed Gulbenkian 2010-06-17T19:28:10Z 2010-06-18T12:00:13Z <p>Let C be a compact subset of the Euclidean plane E whose boundary is a Jordan curve J. If C tiles the plane, can J be such that it has a unique tangent line at each point and none of its sub-arcs is a straight line segment with distinct end-points? If so, can you give an example? J does not need to be convex and the tiling need not be regular. The only requirement is that the plane E be a countable union of congruent copies of C, no two of which have a common interior point.</p> http://mathoverflow.net/questions/28556/a-question-about-tilings-of-the-plane/28559#28559 Answer by Tom Goodwillie for A question about tilings of the plane Tom Goodwillie 2010-06-17T19:44:23Z 2010-06-17T19:44:23Z <p>$C = \lbrace (x,y)| sin\ x\le y\le 2\pi+sin\ x ,\ sin\ y\le x\le 2\pi+sin\ y\rbrace$.</p> http://mathoverflow.net/questions/28556/a-question-about-tilings-of-the-plane/28565#28565 Answer by Joel David Hamkins for A question about tilings of the plane Joel David Hamkins 2010-06-17T20:19:36Z 2010-06-18T01:52:37Z <p>I have a way to do it, but read my note at the bottom about why you may not find it to be valid.</p> <p>Use an eye-shaped figure, but where the ends of the eyes meet at angle 0. This will allow unique tangents at the "corners".</p> <p><img src="http://www.freeimagehosting.net/uploads/1e7dbd258e.jpg" alt="alt text"></p> <p>The eyes will fit together in a standard brick pattern. </p> <p>In the image, I used a sin function, but you can do it also using circle fragments. In this case, the tiling corresponds to a standard penny tiling of the plane, but apportioning the empty space to adjacent pennies in order to make the eye shapes. </p> <p>Since my Jordan curves turn directly around in the opposite direction at those cusps, however, you may not consider this to be a valid example, since perhaps you regard this as two tangent lines at those points, pointing in opposite directions.</p> <p>I think someone will show up and prove that you cannot do it without any cusps. I would like to know whether you can do it with only one cusp.</p>