There is another way to go from a dodecahedron to a rectangle. It's not a projection but a combinatorial mapping with software by J. W. Cannon, W. J. Floyd, and W. R. Parry. http://www.math.vt.edu/people/floyd/research/software/subdiv.html (see bottom of the page for the dodecahedron data) The first program subdivide.c prompts you for a subdivision rules file and a tiling file for a quadrilateral, and returns the tiling file for the subdivision. Once you have the tiling file you can convert it with a second program tilepack.c to use with Ken Stephenson's circle packing program, CirclePack http://www.math.utk.edu/~kens/CirclePack/, to draw the tiling to be Euclidean or hyperbolic. Alternatively you can use the third program squarect.c to create a squared rectangle from the dodecahedron. see http://www.math.vt.edu/people/floyd/research/papers/fsr.pdf for more Stuart Anderson
On Mon, Mar 10, 2014 at 1:58 PM, 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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