  The vertices of the four dimensional $120$-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 $\tau$. The two are related by the conjugation $\tau
\mapsto\tau' = -1/\tau$. 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'
$\Sigma$ which itself turns out to have a natural structure of the unitary
group $SU(2,\mathcal R)$ over the ring $\mathcal R = \mathbb
Z[\frac{1}{2},\tau]$ (called here golden numbers). Acting upon it is the
naturally associated infinite reflection group $H^{\infty}$, which we prove is
of index $2$ in the orthogonal group $O(4,\mathcal R)$. The paper makes
extensive use of the quaternions over $\mathcal R$ and leads to 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.
