# Characterizing and counting the unique vertices in a geodesic sphere based on a triangulated icosahedron

I have a geodesic sphere generated by triangulated an icosahedron to some frequency $\nu$, with a circumscribing sphere of radius $R_c$.

Let's define a 'component' of the geodesic sphere to be a particular vertex and its associated set of edges, and to accommodate this definition, we'll allow different vertices to share the same edges in the graph when necessary. Here, when the length or angles between edges surrounding any two vertices $v_a$ & $v_b$ differ, we'll define $v_a$ & $v_b$ to be 'unique'.

Provided the above definitions, and as a function of $\nu$ and $R_c$, what are the relative populations of 'unique' vertices in my geodesic sphere and is there a simple way to characterize their edge lengths/angles? Is there a simple way to extend these results to the geodesic sphere's dual?

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