[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