Solid geometry questions and musings

Jul 3 03:35 UTC 2003

My less-than-perfect results from guesstimating the angles for
my geodesic panel dome model led me to think about how I could
have gotten it exactly right the first time.  Clearly I have
not studied enough solid geometry.

The first part of solving a problem is stating it correctly.
Looking at the partial model, the completed state would appear
to have twelve pentagonal junctions making it a dodecahedron.
If I wanted to achieve best symmetry, what I think I would want
is to have all the vertices at the same distance from the center.

This approach seems fairly straightforward:  all I have to do is
arrange to have the center vertex of each pentagon and each of the
vertices at the outside tips of each triangle at the same distance
from the center of the dodecahedron.  With a little bit of
trigonometry, the included angle between the 5 triangles of the
pentagon falls out.

This may not be the correct approach, but one trial would tell.
My stumbling block is that I don't have a shortcut for calculating
the distance from the center of the icosahedron to the surface.
(Perhaps with a bit of study of the incomplete model one will
become obvious, but I haven't tried really hard yet.)

So, how do I solve this little problem of solid geometry?  And
what is everyone else contending with, either from a position of
knowledge or (as in my case) ignorance seeking enlightenment?