Penrose Floor

DSCF1546b (C) John Goldsmith

Attempting to tile a plane with regular pentagons must necessarily leave gaps, Roger Penrose found a particular tiling pattern in which the gaps may be filled with three other shapes: a star, a boat, and a diamond.

In addition to the tiles, Penrose stated rules, usually called matching rules, that specify how tiles must be attached to one another; these rules are needed to ensure that the tilings are nonperiodic.
There are three distinct sets of matching rules for pentagonal tiles, shown in different colors in the illustration. This leads to a set of six tiles: a thin rhombus or “diamond”, a five pointed star, a “boat” and three pentagons.