2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78330Lattices and Z-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain 4D examples that are related to 4D root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.24 pages; written for a special issue of Can. J. Math. in honour of H.S.M. CoxeterMetric GeometryMathematical PhysicsNumber Theorytext