How do shapes fill space?
Patch of Penrose tiling. (Image © Edmund Harriss)
Researchers from five institutions are studying how space can be filled with shapes and what this can tell about the natural world and medieval art.
The problem of how shapes fit together has been considered for most of the history of humanity, probably first emerging in the arts. Mathematicians study tilings to help understand symmetry and the structure of crystals. Quasicrystals, ordered solids whose structure never repeats, were only discovered 25 years ago, yet their structure bears remarkable similarity to patterns used in Islamic tiling art, created centuries before.
"Throughout history, artists have used tilings to produce beautiful images," says Dr Edmund Harriss from Imperial College London. "They also have important applications in mathematics and science."
The modern era for tilings began in the 1960s when mathematicians discovered it is impossible to write a computer program that can decide on whether a given set of shapes can tile the plane. It led scientists to create new theories about tilings, including substitution rules for highly ordered structures that never repeat, and discover fascinating examples to back up this theory in Islamic art.
"Tilings are key to understanding how space can be organised, for example the symmetries that can occur," continues Edmund.