  The purpose of this article is to demonstrate that non-crystallographic
reflection groups can be used to build new solvable quantum particle systems.
We explicitly construct a one-parametric family of solvable four-body systems
on a line, related to the symmetry of a regular icosahedron: in two distinct
limiting cases the system is constrained to a half-line. We repeat the program
for a 600-cell, a four-dimensional generalization of the regular
three-dimensional icosahedron.
