This work is related to the study of pattern formation in symmetric physical systems. Our purpose is to discuss a possible model, namely the projection model, to explain the appearance and evolution of regular patterns in symmetric systems of equations.

  Results found in Crystallography and Equivariant Bifurcation Theory are used extensively in our work. In particular, we provide a formalism of how the model of projection can be used and interpreted to understand experiments of reaction-diffusion systems.

We construct a scenario where systems of symmetric PDEs posed in different dimensions can be compared as projection. In particular, we show how we can overcome the boundary conditions imposed by the problems.

We prove a correspondence between irreducible representations and fixed points subspaces, given by the action of a (n+1)-dimensional crystallographic group, with the action of its projection on lower dimension. Such results are the first step to compare typical structures in dimension (n+1), after projection, and the typical solutions of the posed problem in dimension n. 

We show that complex structures, as the black-eye pattern, obtained both as projection and as an experimental observation in CIMA reactions are the same. In particular, we believe that the projection model provides extra information to the study of pattern forming system, since it allows us to embed the original problem into one with more symmetry.