  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 the Kac-Moody-type extensions
considered in Dechant et al. 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.
