[math-fun] Projections -> sphere <-> rectangle
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
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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P.S. Here’s a less-technical discussion of the same paper: < http://mathtourist.blogspot.com/2007/06/wrapping-perfect-sphere.html >. —Dan On Mar 10, 2014, at 11:21 AM, Dan Asimov <dasimov@earthlink.net> wrote:
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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Hilarie, Ron Hardin and I looked at a related problem years ago. Fix N, say ... 24 or 25 or ... 100 or ... Place N points on a sphere so that the volume of the convex hull is maximized. There are numerical tables produced by extensive computations that you can find on my home page http://neilsloane.com - see under Tables. It is a kind of dual to your question. Neil 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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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
You might also be interested by this lecture, "La pixellisation de la sphere" https://www.youtube.com/watch?v=7vg2s262khc recently given at IHP for the Henon days. The lecture is in french but the slides are in english. On Mon, Mar 10, 2014 at 8:02 PM, Neil Sloane <njasloane@gmail.com> wrote:
Hilarie, Ron Hardin and I looked at a related problem years ago. Fix N, say ... 24 or 25 or ... 100 or ... Place N points on a sphere so that the volume of the convex hull is maximized. There are numerical tables produced by extensive computations that you can find on my home page http://neilsloane.com - see under Tables. It is a kind of dual to your question. Neil
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Dan Asimov -
Hilarie Orman -
Neil Sloane -
Olivier Gerard