Benjamin McCabe with Rolfe Petschek

Animating Tessellations of the Plane with Statistical Models

(started in spring 2009 and will be resumed and completed in spring 2010)

Artists, particularly M.C. Escher, have made many attractive tessellations of the plane, incidentally illustrating many of the two dimensional space groups. Such pictures may have different space groups or attractiveness based on the patterning and symmetry of various colors in the pictures, and in defects in such symmetries. Many physical systems have “broken” symmetries in which an unspecified state (e.g. color) can be determined by the state of its neighbors, as in antiferromagnetism. Much is known about such ordering in various two dimensional cases, including various statistical models thereof. In this project we will combine art and physics by animating an Escher-like tessellation of the plane, using a Monte-Carlo or similar simulation of a statistical model to change the colors or other attributes of the tessellation. As the formation of long range order in such models, and the resulting potential fractional dimensionality of the patterns is thought to have aesthetic value, this will hopefully achieve all of providing an interesting illustration of two dimensional statistical models, an interesting piece of modifiable art, and a potential screen-saver.