This paper, “Wrapping spheres with flat paper”, may relate to your question: < http://erikdemaine.org/papers/SphereWrapping_CGTA/paper.pdf > —Dan On Mar 10, 2014, at 10:58 AM, Hilarie Orman <ho@alum.mit.edu> wrote:
A while back I asked if there was illustration of a project of a dodecahedron to a sphere to a rectangle. Once reasonable way to do the calculation is to compute the vertices on the sphere (a nice collection of 1, phi, and 1/phi for the coordinates), the great circle arcs between the vertices, and to use a Miller projection of that to a rectangle. I've not found a picture of this for a dodecahedron, though. The Internet has pictures of almost every other conceivable representation of a dodecadhedron, just not this one.
A somewhat related question is how would you shrink wrap a ping-pong ball? A rectangle of ordinary kitchen clear plastic wrap stretches nicely across a hemisphere, but the lower part bunches up along longitudinal lines and creates a dense knot at the pole. How would you even describe a solution that minimizes the surface area, short of saying "use a spherical layer of plastic wrap"?
Hilarie
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