On the Number of Tubes Touching a Sphere or a Tube
A problem is formulated about how many unit-radius tubes can touch a ball of given radius from the outside and from the inside.
Upper bounds for the maximum numbers of contacts are obtained for both interior and exterior contacts. It is also shown that
the maximum number of unit-radius tubes touching the same orthogonal cross-section of a particular tube of radius P is [π (arcsin(P+1)−1)−1] and if the number of contacts takes on its maximum, then all tubes are locally aligned.