Discretization of SU(2) and the orthogonal group using icosahedral symmetries and the golden numbers

Abstract

ABSTRACTThe vertices of the four-dimensional 600-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group H4. There are two special coordinate representations of this root system in which they and their corresponding Coxeter groups involve only rational numbers and the golden ratio τ. The two are related by the conjugation τ↦τ′ = −1∕τ. This paper investigates what happens when the two root systems are combined and the group generated by both versions of H4 is allowed to operate on them. The result is a new, but infinite, ‘root system’ Σ which itself turns out to have a natural structure of the unitary group SU(2,ℛ) over the ring (called here golden numbers). Acting upon it is the naturally associated infinite reflection group H∞, which we prove is of index 2 in the orthogonal group O(4,ℛ). The paper makes extensive use of the quaternions over ℛ and leads to a highly structured discretized filtration of SU(2). We use this to offer a simple and effective way to approximate any element of SU(2) to any degree of accuracy required using the repeated actions of just five fixed reflections, a process that may find application in computational methods in quantum mechanics.