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The purpose of this article is to demonstrate that non-crystallographicreflection groups can be used to build new solvable quantum particle systems.We explicitly construct a one-parametric family of solvable four-body systemson a line, related to the symmetry of a regular icosahedron: in two distinctlimiting cases the system is constrained to a half-line. We repeat the programfor a 600-cell, a four-dimensional generalization of the regularthree-dimensional icosahedron.

Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra