Penrose tiling

Example

A Penrose tiling is a tiling of the plane by copies of two prototiles that obey matching rules. Roger Penrose introduced these tilings in the 1970s. In the rhombus version, the prototiles are two rhombi with sides of length $1$. The thick rhombus has angles $72^\circ$ and $108^\circ$, and the thin rhombus has angles $36^\circ$ and $144^\circ$. The matching rules mark the edges of the rhombi and allow two tiles to meet along an edge only when the marks agree. In the kite and dart version, the two prototiles are a convex kite and a non-convex dart, again with matching rules.

The two prototiles with their matching rules admit tilings of the whole plane, and none of these tilings is periodic. So each version is an aperiodic set of prototiles. In every Penrose tiling the ratio of the number of thick to thin rhombi, counted in large disks, tends to the golden ratio $\varphi = (1 + \sqrt{5})/2$. Since $\varphi$ is irrational, this ratio cannot come from a periodic tiling, whose ratio would be the rational ratio within one period.