Here's The Maths Behind Those 'Impossible' Never-Repeating Patterns

Remember the graph paper you used at school - the sort covered in tiny squares?

It’s a neat way to picture what mathematicians describe as a periodic tiling of space, where shapes cover an area completely without any overlaps or gaps.

If we shift the entire pattern by exactly one tile length (translate it), or rotate it by 90 degrees, we end up with the very same arrangement. That happens because, in this example, the tiling as a whole shares the same symmetry as the individual tile.

Now picture trying to tile a bathroom using pentagons instead of squares. That won’t work: the pentagons cannot meet edge-to-edge everywhere without either leaving gaps or overlapping.

Patterns (built from tiles) and crystals (built from atoms or molecules) are usually periodic in the same way as graph paper, and their symmetries are closely related.

Out of all the possible ways atoms could arrange themselves, nature tends to favour these regular, repeating structures because they correspond to the lowest energy cost of assembly. In fact, it’s only been for the past couple of decades that we’ve known crystals can also exist with non-periodic tiling - patterns that never repeat.

My colleagues and I have now developed a model that helps explain how this can happen.

Penrose patterns and five-fold symmetry

Back in the 1970s, the physicist Roger Penrose showed that a pattern could be created from two distinct shapes whose angles and sides are linked to those of a pentagon. This arrangement looks unchanged when rotated by 72-degree angles - so over a full 360-degree turn, it matches itself from five different orientations.

Within this pattern, many small fragments appear again and again. For instance, in the graphic referred to below, the orange five-pointed star turns up repeatedly.

Yet each of those stars is bordered by a different surrounding configuration. That means the overall pattern never repeats in any direction. So the graphic is an example of a pattern with rotational symmetry but no translational symmetry.

From tiling to 3D quasicrystals

Moving to three dimensions makes the story more intricate. In the 1980s, Dan Schechtman studied an aluminium–manganese alloy that displayed a non-periodic pattern in every direction, while still exhibiting rotational symmetry under the same 72-degree rotation.

Up to that point, crystals without translational symmetry but with rotational symmetry had essentially been unthinkable - and many scientists rejected the observation. It became one of those rare moments when the definition of "What is a crystal" had to be revised to accommodate a new discovery. As a result, these materials are now known as ‘quasicrystals’.

Irrational number

The non-repeating nature of a quasicrystal pattern comes from an irrational number at the core of how it is built. In a regular pentagon, the ratio between the side length of the inscribed five-pointed star and the side length of the pentagon itself is the famous irrational number ‘phi’ (not to be confused with pi), which is about 1.618.

This value is also called the golden ratio (and it also satisfies the relation phi = 1+1/phi). As a consequence, when a quasicrystal is assembled from tiles derived from a pentagon - like those Penrose used - we see rotational symmetry at 72-degree angles.

This five-fold symmetry shows up both in the quasicrystal image as ten radial lines around a central red dot (above), and in a scale model of the quasicrystal’s central region constructed using Zometool (below).

In the model, it’s helpful to treat the white balls as the sites where the particles/atoms of the crystal structure would lie, and to view the red and yellow rods as bonds between particles, indicating the shapes and symmetries within the structure.

A model for 3D quasicrystals

In our recent publication, we set out two characteristics a system must possess to produce a 3D quasicrystal. First, the system must contain patterns at two distinct sizes (length-scale) that appear in an appropriate irrational ratio (such as phi).

Second, these two length-scales must be able to interact strongly with one another. Alongside never-repeating quasicrystal patterns, the same model can also generate other familiar, regular crystal structures, including hexagons, body-centred cubes and so on.

A model like this allows us to investigate how these competing patterns vie with one another, and to pinpoint the conditions under which quasicrystals will form in nature.

The mathematical framework for producing never-repeating patterns is extremely valuable: it helps us understand how such structures arise and even how to design them with chosen properties. That is why we at the University of Leeds, together with colleagues at other institutions, are so interested in researching these questions.

However, this work isn’t merely an abstract mathematical concept (even if the mathematics behind it is rather addictive) - it also has strong potential for practical uses, including highly efficient quasicrystal lasers. The reason is that when periodic crystal patterns are used in a laser, the symmetry of the repeating structure produces a low-power laser beam.

By introducing defects into the crystal pattern, or instead using a never-repeating quasicrystal pattern at the output end of a laser, it becomes possible to create an efficient laser beam with high peak output power.

Elsewhere, some researchers are even looking at whether quasicrystals could produce reflective finishes if they were added to household paint.

Priya Subramanian, Research Fellow Applied Mathematics, University of Leeds.

This article was originally published by The Conversation. Read the original article.