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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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Perhaps all the vertices within one-third of an icosahedron face are unique? It depends on what are known as the "chord factors" for each geodesic segment, which vary systematically over each icosahedron face. Not absolutely certain of this...

However, you can find out by examining Rick Bono's DOME program, which is all over the web, and apparently the source of nearly every geodesic dome created in graphics. E.g., try here or here. This is an image (by "Earl") taken from the latter site:
           enter image description here

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