A fairly recent branch of recreational mathematics, non-periodic/aperiodic tiling took off as a serious branch of materials science when Danny Schectman discovered metal alloys displaying a flagrant 5-fold symmetry in their X-ray diffraction patterns.
Roger Penrose discovered several versions of a particular 2D tiling that came to be know as Penrose tiles, and Ammann and others furthered our understanding of the nature of such irrational "tesselations". Schectmannites (the aforementioned alloys) are basically in a structure that is a 3D extension of the Penrose tiling structure.
There are strong fractal properties of many of these tilings, in that a tiling can be transformed into a denser tiling by simple substitution rules, and this repeated indefinitely - a processes termed deflation and inflation.
Most of these links go to the wonderful University of Minnesota's Geometry Centre pages - a well nice net resource.