In that linked MathOverflow post, I find this question extremely interesting: "Within the configuration space, is a continuous transition possible from the configuration in Figures 1 and 2 to the configuration in Figure 3?" I feel the answer must be No!, but I don't see how to prove it yet. If a cylinder is identified with the space it occupies, then my guess is that the only continuous deformation of the configuration shown in the rightmost picture of Figure 3 are those achieved by rigid rotations of the whole shebang. --Dan On Oct 12, 2014, at 3:32 PM, Veit Elser <ve10@cornell.edu> wrote:
In an answer to a related question on MO,
http://mathoverflow.net/questions/156008/how-many-unit-cylinders-can-touch-a...
there is a claim that the parallel cylinder configuration can already be beat for k = 6. The sphere radius is given as approximately 0.9527. However, there is no proof or coordinates, just an unconvincing graphic.
-Veit
On Oct 12, 2014, at 10:20 AM, Bill Gosper <billgosper@gmail.com> wrote:
On 2014-10-12 05:57, Adam P. Goucher wrote:
Veit,
For k = 6 cylinders, there's a pyritohedral configuration which achieves precisely the same radius as the parallel hexagonal configuration (namely r(k) = 1):
https://twitter.com/wolframtap/status/521283214552616960
I conjecture that these are the only optimal configurations for k = 6, up to isometries.
Sincerely,
Adam P. Goucher
Sent: Sunday, October 12, 2014 at 1:01 PM From: "Veit Elser" <ve10@cornell.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Tammes cylinders
Pack k unit-radius, infinite-length cylinders so they all are tangent to the same sphere.
Now minimize the radius of the sphere. Call this minimum radius r(k).
For small k the minimum is achieved with parallel cylinders, and r(k) = 1/sin(pi/k) - 1.
For large k one can do better. The smallest k I’ve found, that beats the parallel packing, is k = 12.
Can you find a packing for smaller k that also beats the parallel cylinder upper bound?
-Veit
Do eight cylinders collide if laid on the faces of an octahedron? --rwg Also, Adam should probably have told us about http://conwaylife.com/wiki/Parallel_HBK a large spaceship that actually drags itself along (with minute velocity.) at slope 2, vs the huge http://conwaylife.com/wiki/Gemini which copies and deletes itself at slope 5. The latter's huge genome is encoded in the spacings of collinear gliders. The HBK (Half-Baked Knightship)'s genome is encoded in the positions of thousands of half-bakeries (twinned loaves).
How long before we see a natural clump of a few dozen dots moving with slope 2? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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