Roger Penrose explains the mathematics of the Penrose Paving

As you enter the new Mathematical Institute here in Oxford you are confronted with a pattern of beauty and intrigue. Designed by Sir Roger Penrose, Emeritus Rouse Ball Professor of Mathematics, and expanding on his discovery of almost thirty years ago, the Penrose Paving is constructed from just two different diamond-shaped granite tiles, each adorned identically with stainless steel circular arcs. There are various ways of covering the infinite plane with them, matching the arcs. But every such pattern is non-repetitive and contains infinitely many exact copies of what you see before you. 

 

A longer version, expanding on the mathematics is also available