Re: [math-fun] Upending the sphere — the discrete version: Puzzle
Closely related questions are the same but for rolling a polyhedron on another one with the same faces: upending an octahedron or an icosahedron, rolling on a tetrahedron, or on an octahedron, or on an icosahedron. —Dan
I am pretty sure the cuboctohedron can be done, though.
On Wed, Nov 7, 2018 at 7:24 PM Allan Wechsler <acwacw@gmail.com> wrote:
The rhombic dodecahedron also can't be done, but for entirely different reasons. I think.
On Wed, Nov 7, 2018 at 7:17 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, Nov 7, 2018 at 5:10 PM Allan Wechsler <acwacw@gmail.com> wrote:
Wait, I don't have an answer for the octohedron, and suspect it can't be done.
I think you're right, since you can color both the octahedron and the triangular tiling with two colors such that any move also changes the color of the tangent faces.
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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Two spheres of different radii? On Wed, Nov 7, 2018 at 8:25 PM Dan Asimov <dasimov@earthlink.net> wrote:
Closely related questions are the same but for rolling a polyhedron on another one with the same faces: upending an octahedron or an icosahedron, rolling on a tetrahedron, or on an octahedron, or on an icosahedron.
—Dan
I am pretty sure the cuboctohedron can be done, though.
On Wed, Nov 7, 2018 at 7:24 PM Allan Wechsler <acwacw@gmail.com> wrote:
The rhombic dodecahedron also can't be done, but for entirely different reasons. I think.
On Wed, Nov 7, 2018 at 7:17 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, Nov 7, 2018 at 5:10 PM Allan Wechsler <acwacw@gmail.com> wrote:
Wait, I don't have an answer for the octohedron, and suspect it
can't be
done.
I think you're right, since you can color both the octahedron and the triangular tiling with two colors such that any move also changes the color of the tangent faces.
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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This conversation reminds me of an old project of mine: trying to devise a three-dimensional version of the rolling-polygons story. (See for instance https://www.math.hmc.edu/funfacts/ffiles/20007.2-3.shtml .) I'm pretty sure I discussed the idea in this forum. I never found any interesting results of this kind, but I haven't completely given up hope. As I recall, one of you suggested that we roll a cube not on a flat table top but on a stepped "Q*bert surface"; I can't remember whether that led to anything. Jim Propp On Wed, Nov 7, 2018 at 10:32 PM Allan Wechsler <acwacw@gmail.com> wrote:
Two spheres of different radii?
On Wed, Nov 7, 2018 at 8:25 PM Dan Asimov <dasimov@earthlink.net> wrote:
Closely related questions are the same but for rolling a polyhedron on another one with the same faces: upending an octahedron or an icosahedron, rolling on a tetrahedron, or on an octahedron, or on an icosahedron.
—Dan
I am pretty sure the cuboctohedron can be done, though.
On Wed, Nov 7, 2018 at 7:24 PM Allan Wechsler <acwacw@gmail.com> wrote:
The rhombic dodecahedron also can't be done, but for entirely different reasons. I think.
On Wed, Nov 7, 2018 at 7:17 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, Nov 7, 2018 at 5:10 PM Allan Wechsler <acwacw@gmail.com> wrote:
Wait, I don't have an answer for the octohedron, and suspect it
can't be
done.
I think you're right, since you can color both the octahedron and the triangular tiling with two colors such that any move also changes the color of the tangent faces.
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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participants (3)
-
Allan Wechsler -
Dan Asimov -
James Propp