Golden Section
Penrose Tilings and the Golden Mean

The British physicist and mathematician, Roger Penrose, has developed an aperiodic tiling which incorporates the golden section and the five-fold symmetry inherent in it. The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B).

diagram

When a plane is tiled according to Penrose's directions, (described in The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics , Oxford University Press, 1989, and also in Tilings and Patterns by Grunbaum and Shephard) the ratio of tile A to tile B is the Golden Ratio.

Penrose Tilings

By following sets of parallel lines within the tiling, the five-fold symmetry is revealed. I have emphasized two of the five axes in the images below.

The Symmetry of Penrose TilingsThe Symmetry of Penrose Tilings

In addition to the unusual symmetry, Penrose tilings reveal a pattern of overlapping decagons. Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean. I have highlighted the two types of decagons below.

Decagons in Penrose Tilings

In the years since Roger Penrose developed the first Penrose tiling, other scientists and Penrose himself have continued to develop new aperiodic tilings. In the February1992 issue of Scientific American Peter W. Stephens and Alan I. Goldman reported that quasicrystals, alloys formed by melting aluminum, copper, and iron together, often revealed the same symmetry present in Penrose Tilings.

Original webpage by Rashomon