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?
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