{"id":78,"date":"2013-08-15T15:20:52","date_gmt":"2013-08-15T15:20:52","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=78"},"modified":"2015-07-29T00:55:36","modified_gmt":"2015-07-29T00:55:36","slug":"tiling","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/08\/15\/tiling\/","title":{"rendered":"T\u00fcbingen Tiling"},"content":{"rendered":"
\"T\u00fcbingen<\/a>

T\u00fcbingen Tiling – Greg Egan<\/p><\/div>\n

A systematic way to generate quasiperiodic tilings of the plane is to take a lattice in higher dimensions and slice it at a funny angle.\u00a0 Greg Egan’s T\u00fcbingen<\/a> applet generates quasiperiodic tilings by projecting selected triangles from an $n$-dimensional lattice called the $\\mathrm{A}_n$ lattice onto a plane. This particular picture comes from the $\\mathrm{A}_4$ lattice. The applet produces moving pictures<\/i> that are much more beautiful than this still image, so please check it out!<\/p>\n

The $\\mathrm{A}_n$ lattice lives in $n$ dimensions, but it’s easiest to describe it in one more dimension, as the set of all $(n+1)$-tuples of integers $(x_1,…,x_{n+1})$ such that<\/p>\n

$$ x_1 + \\cdots + x_{n+1} = 0. $$<\/p>\n

It’s a fun exercise to show that $\\mathrm{A}_2$ is a 2-dimensional hexagonal lattice, the sort of lattice you use to pack pennies as densely as possible. Similarly, $\\mathrm{A}_3$ gives a standard way of packing grapefruit, which is in fact the densest lattice packing of spheres in 3 dimensions. If you were stacking layers of 4-dimensional grapefruit you could use the $\\mathrm{A}_4$ lattice, though that would not be the densest possible packing.<\/p>\n

Let me rapidly sketch how we get from the $\\mathrm{A}_4$ lattice to the beautiful tiling shown here.<\/p>\n

Each point $x$ in the $\\mathrm{A}_4$ lattice is surrounded by a Voronoi cell<\/a>, which consists of all points that are closer to $x$ than to any other lattice point. The Voronoi cells of $\\mathrm{A}_4$ are all identical convex polytopes—can you figure out what this polytope is?<\/p>\n

The cells dual to these Voronoi cells are called Delaunay cells<\/a>. To get the tiling we pick a plane $P$ in 4 dimensions, and whenever $P$ intersects a 2-dimensional face of a Voronoi cell, we project the corresponding 2d face of the corresponding Delaunay cell, which is a triangle, onto $P$. Then we draw these triangles on the plane!<\/p>\n

For more details read:<\/p>\n