vivid writes, above:
[Any geodesic sphere has] only 12 vertices where 5 triangles come together. All the other vertices have 6 triangles meeting.
This is a bit
odd, surely.
Why have exactly
12 vertices of
degree 5, and all others of degree
6? Wasn't
Buckminster Fuller smart enough to manage to change this?
Actually, it's not Fuller's fault. He couldn't do anything about it; it's mathematics that gets in the way!
Let's set the scenario: we're constructing a "sphere-like" polytope in R3 out of triangles. We want each vertex to have degree 5 or 6.
Sounds like a job for the Euler characteristic, V-E+F=2! Let's call the number of vertices V, the number of edges E, and the number of faces F. Also, let's say we have n vertices of degree 5 (and consequently V-n of degree 6). Every edge participates in 2 triangles, so 3*F=2*E. Every edge has 2 vertices on it, so we have 2*E=5*n+6*(V-n), so 3F=6V-n. And V-E+F=2.
Turns out we can solve for n:
6V-6E+6F=12
(3F+n) - (9F) + 6F = 12
n = 12
So
Euler say s we've got to have exactly 12 vertices of degree 5 in this situation.