According to my math, and http://en.wikipedia.org/wiki/Icosahedron,
for a sphere of outer radius R you need triangles with edge length 1.323*R
and bevel angle 20.905 degrees.

The minimum thickness of the wood must be 0.26 * R, any extra thickness
becomes the thickness of the resulting sphere.

Thus, for a sphere of radius 3 7/8, you want triangles with edges
5.127" long and wood (1+T)" thick (T = thickness of sphere wall).