Radial tilings with variable area ratios - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:51:44Z http://mathoverflow.net/feeds/question/83145 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83145/radial-tilings-with-variable-area-ratios Radial tilings with variable area ratios Joel Ford 2011-12-10T21:11:48Z 2011-12-11T02:10:27Z <p>I was looking at this neat page on logarithmic spiral tilings when a question popped up:</p> <p><a href="http://www.uwgb.edu/dutchs/symmetry/log-spir.htm" rel="nofollow">http://www.uwgb.edu/dutchs/symmetry/log-spir.htm</a></p> <p>It seems that in all of the tilings shown, the area of each tile is exponentially increasing as a function of the distance to the origin. Are there any radial- or spiral-type tilings (or "tiling-like" configurations) in which the area of each tile is a <em>polynomial</em> function of the distance to the origin, say \$r^{1/2}\$ or \$r^2\$? I don't require similarity of shapes or anything, just a simple way to fill the plane with shapes possessing (or approximately possessing) this property. I guess it would be nice if the shapes stay reasonable rounded and convex (i.e. not really long and skinny).</p> http://mathoverflow.net/questions/83145/radial-tilings-with-variable-area-ratios/83148#83148 Answer by Joseph O'Rourke for Radial tilings with variable area ratios Joseph O'Rourke 2011-12-10T21:57:02Z 2011-12-11T02:10:27Z <p><br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="http://people.csail.mit.edu/~orourke/MathOverflow/TilingsAndPatterns.jpg" alt="Grunbaum and Shephard book cover"><br /> See Section 9.5, "Spiral Tilings," p.512ff. These remarkable tilings go back to H. Voderberg in the 1930's.</p>