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The vertices of the four dimensional $120$-cell form a non-crystallographicroot system whose corresponding symmetry group is the Coxeter group $H_{4}$.There are two special coordinate representations of this root system in whichthey and their corresponding Coxeter groups involve only rational numbers andthe golden ratio $\tau$. The two are related by the conjugation $\tau\mapsto\tau' = -1/\tau$. This paper investigates what happens when the two rootsystems are combined and the group generated by both versions of $H_{4}$ isallowed to operate on them. The result is a new, but infinite, `root system'$\Sigma$ which itself turns out to have a natural structure of the unitarygroup $SU(2,\mathcal R)$ over the ring $\mathcal R = \mathbbZ[\frac{1}{2},\tau]$ (called here golden numbers). Acting upon it is thenaturally associated infinite reflection group $H^{\infty}$, which we prove isof index $2$ in the orthogonal group $O(4,\mathcal R)$. The paper makesextensive use of the quaternions over $\mathcal R$ and leads to highlystructured discretized filtration of $SU(2)$. We use this to offer a simple andeffective way to approximate any element of $SU(2)$ to any degree of accuracyrequired using the repeated actions of just five fixed reflections, a processthat may find application in computational methods in quantum mechanics.

Discretization of SU(2) and the Orthogonal Group Using Icosahedral Symmetries and the Golden Numbers