Brilliant Logo Triangles

The logo is based on the disdyakis triacontahedron, which forms 15 great circles when projected onto a sphere, exhibiting both icosahedral and dodecahedral symmetries.

Any 3 points on the sphere which satisfy the following conditions

1) Each point is the intersection of 2, 3, or 5 of the great circles
2) Each pair of points lie on 1 of the great circles
3) Not all three points lie on any 1 of the great circles

shall be counted as 1 [spherical] triangle. How many triangles are there in the spherical disdyakis triacontahedron?

Below is the stereographic projection of the spherical disdyakis triacontahedron. One of the points is at infinity.

Note: The disdyakis triacontahedron is the largest Catalan solid with 120 identical faces, which is the maximum any polyhedron can have.

Follow-up notice: Some have questioned about the fact there can be more than one possible spherical triangle, given 3 arbitrary points on a sphere. That is the reason why it's been stated that any such unordered set of 3 points meeting the conditions 1-3 shall count as ONE spherical triangle. Hence, it becomes more of a problem in combinatorics rather than geometrical, although a good understanding of geometry helps.

2nd Follow-up notice: \(\dbinom{62}3=37820\) is not the answer, as it would violate conditions 2) and 3). The great circles are limited to the 15 noted.


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