Discretization of SU(2) and the orthogonal group using icosahedral symmetries and the golden numbers. Issue 6 (3rd June 2018) - Moody, Robert V.; Morita, Jun
ABSTRACT: The vertices of the four-dimensional 600-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group H 4 . 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 H 4 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 ℛ = ℤ [ 1 2, τ ] (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.