Aperiodic Tessellations

From EscherMath

So far we have explored tessellations, which are periodic. They look the same at different points of the plane. It is possible however to create tessellations that are not periodic. Their patterns do not quite repeat throughout the plane. We will discuss two different ways of creating aperiodic tessellations.

Tilings with no translation symmetry are called aperiodic tilings.

Random Tessellations

An easy method for creating an aperiodic tiling is to take two squares and decorate them in different ways. For example:

We label the squares with “0” and “1”. A random number generator can generate a string of 0’s and 1’s. Random implies that there is no pattern what so ever, so that creating a tessellation from these random numbers will result in a tessellation with no pattern to it. Using a random number generator on the web ([[1]] for instance) one can get the following 8×8 grid which leads to the following tessellation:

Penrose Tilings

Johannes Kepler was one of the first scientists to experiment with star polygons, and arrive at some interesting tessellations. It seems that Kepler experimented with shapes that have 5-fold symmetry.

The later study of truly aperiodic tilings was strongly influenced by Roger Penrose (he later became SIR Roger Penrose) An excellent website showing Kepler’s tessellations as well as some of Penrose’s tilings can be found at: [[2]]

The tessellations most commonly known as Penrose tilings are made up of kites and darts. They should be glued together so that the marked vertices line up.

Below are some images of aperiodic penrose tilings. There are infinitely many aperiodic tilings, and it is also true that every finite portion of any tiling is contained in every other tiling. This actually implies that a small patch of tiles does not determine the tessellation. This is very different from the periodic tilings, where a small segment of the tessellation determines the entire pattern.