This is a research activity only at a tool level, but a research interest at all levels

The symmetries appearing in a system are partially inherited by the variables which describe it. For example, when water waves appear in a relatively small container, they, on one hand, follow, at least partially, the shape of the container. On the other hand, the waves could be, or not, periodic in one of the spatial directions, because of the spatial translations invariance of the governing equations.

Actually, the variables verifying a set of symmetries give rise to the structures (in the previous example, the surface displacement as a function of time and position, with its given symmetries). These structures, when their typical lengths and times are at a human scale, are commonly called patterns.

In this sense, one could say that the symmetries in a system are one of the most important issues in order to understand how does it work, or even more important, why does it work. As in Physics many systems can be modelled by (systems) of partial differential equations (or similar continuous mathematical models), it is a major challenge to understand the behavior of them regarding to their symmetry properties.