Certainly this Wikipedia article would yield an answer to the first question: https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions. On Thu, Jun 25, 2020 at 12:25 PM Dan Asimov <dasimov@earthlink.net> wrote:

Suppose ∆ is a spherical triangle on the 2-sphere S^2 with its angles A, B, C each equal to a rational multiple of π.

Now suppose we reflect ∆ across each of its sides, and reflect each of the resulting triangles about each of its sides, etc., indefinitely until there remain no sides that haven't been reflected across.

Let V be the set of all the vertices resulting from these reflections.

Questions:

* For which rational multiples {A, B, C} of π will the set V be finite?

* If V is infinite, does that imply V is dense in S^2 ?

—Dan

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