Affine extensions of non-crystallographic Coxeter groups induced by projection

In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E_8, D_6 and A_4. We show that the induced affine extensions of the non-crystallographic groups H_4, H_3 and H_2 correspond to a distinguished subset of those considered in [P.-P. Dechant, C. Boehm, and R. Twarock, J. Phys. A: Math. Theor. 45, 285202 (2012)]. This class of extensions was motivated by physical applications in icosahedral systems in biology (viruses), physics (quasicrystals) and chemistry (fullerenes). By connecting these here to extensions of E_8, D_6 and A_4, we place them into the broader context of crystallographic lattices such as E_8, suggesting their potential for applications in high energy physics, integrable systems and modular form theory. By inverting the projection, we make the case for admitting different number fields in the Cartan matrix, which could open up enticing possibilities in hyperbolic geometry and rational conformal field theory.