Applications of the icosahedral equation for the Rogers-Ramanujan continued fraction

Let R(q) denote the Rogers-Ramanujan continued fraction for |q| < 1. By applying the RootApproximant command in the Wolfram language to expressions involving the theta function f(-q) := (q;q)_∞ given in modular relations due to Yi, this provides a systematic way of obtaining experimentally discovered evaluations for R(e^-π√(r)), for r ∈ℚ_> 0. We succeed in applying this approach to obtain explicit closed forms, in terms of radicals over ℚ, for the Rogers-Ramanujan continued fraction that have not previously been discovered or proved. We prove our closed forms using the icosahedral equation for R together with closed forms for and modular relations associated with Ramanujan's G- and g-functions. An especially remarkable closed form that we introduce and prove is for R( e^-π√(48/5)), in view of the computational difficulties surrounding the application of an order-25 modular relation in the evaluation of G_48/5.