geometry.college, geometry.forum, Tue, 18 Jan 1994
Penrose tilings are a beautiful nonperiodic tilings of the plane using only two different rhombi. They have a variety of applications. For example, they arise in the study of quasicrystals; see Evelyn Sander, "Quasicrystals," geometry.college, November 15, 1993. The tilings have a rich mathematical structure: Despite their nonperiodicity, they exhibit five-fold symmetry, obey certain strict matching rules, and only allow for a finite number of different ways to put the two tiles together. They also have the property of quasiperiodicity; namely, given any pattern in the tiling, there is an R>0 such that it is possible to find a repeat of the pattern within any ball of radius R.
In 1982 N.G. de Bruijn showed that all Penrose tilings of the plane result from the projection of part of a five-dimensional lattice onto a plane. In fact, by specifying the "slope" and "offset" in five dimensions of the plane upon which to project, one can specify a tiling. If this plane is invariant under five-fold rotation about the body diagonal of the unit cube, and if we chose carefully which part of the lattice is projected, then the tiling is a Penrose tiling.
More specifically, we start with the five-dimensional integer lattice and its connecting 2-facets. Pick a plane E upon which we will project facets and points from the lattice to create the tiling. We choose E orthogonal to the line through the body diagonal of the unit cube. Here is how we decide which are the allowed vertices in the tiling of E: Project the unit cube into the three-dimensional orthogonal complement of E. The resulting object is a rhombic icosahedron which we call K. The allowed vertices are exactly those which land inside K under projection, and the allowed facets are those for which all four associated vertices are allowed. Our restrictions on E and allowed vertices forbid overlaps and gaps in the tiling.
Eugenio Durand, a Geometry Center programmer, has written the program QuasiTiler to find the described quasiperiodic tilings of the plane. He originally wrote it to help Marjorie Senechal with her work on quasicrystals. The program allows the user to specify the "slope" of the plane E, using a mouse to modify a picture of the five-dimensional unit cube. There are three degrees of freedom for the offset of the plane. The user uses three sliders to change the offset. One of the offset directions specifies whether the tiling is a Penrose tiling. The program then shows the tiling that the user has specified. In addition, the user may specify a lattice dimension other than five. It is an easy program to use, and the results are beautiful.
QuasiTiler runs on Next machines. It is available by anonymous ftp in the /u/ftp/pub directory of ftp.geom.umn.edu. Try the WWW version of Quasitiler.
The following reference describes Penrose tiles and briefly mentions the approach used in QuasiTiler:
Ivars Peterson, "The mathematical tourist: snapshots of modern mathematics," New York, Freeman, 1988.
Created: January 3 1995 --- Last modified: Jun 18 1996