Roger Penrose devised a tile, which will tile the hyperbolic (nonEuclidean) plane but will only do so "aperiodically." To describe it, you need to know about the halfspace model of the hyperbolic plane H, http://en.wikipedia.org/wiki/Poincare_half-plane_model i.e. H is represented by the Euclidean upper halfplane y>0, using the distance function dist = arccosh( 1 + ((x1-x2)^2 + (y1-y2)^2) / (2*y1*y2) ). The x-axis plus point at infinity together are the "horocircle at infinity" and translation in x-direction, and rescaling, both are isometries. OK, the tile is described Euclideanly as a unit square with decorated edges. Left edge: small circular knob, fitting into small circular matching hole on Right edge. Top edge: Dilbert-shaped protrusion of size 2*epsilon at center of edge. Bottom edge: two Dilbert-shaped intrusions of size 1*epsilon at the 1/4 and 3/4 points along edge. Now to tile, 1. place a line of tiles with all lower edges along the Euclidean line y=1, tiling, basically, the strip 1<y<2. One of the tiles has left edge on y-axis. 2. scale that line of tiles by factors S that are powers of 2 (...1/8, 1/4, 1/2, 1, 2, 4, 8,...), that is apply the map (x,y) --> S*(x,y). The result is an "Escher" hierarchical tiling of the hyperbolic plane by pseudo-square tiles which seems obviously unique. I'm not sure whether we should count this tiling as "aperiodic." I mean, this tiling has an infinite symmetry group (the scalings by powers of 2). However, it is certainly less symmetric than, e.g, the tiling of the hyperbolic plane by regular 7-gons, with size chosen so that each angle is 120 degrees, which has a oriented-tile-transitive symmetry group. Penrose's tiling in contrast has an infinite number of different "tile types" i.e, there is no symmetry of the tiling which will map a tile in type A to one in type B.