Cylinder packing is the arrangement of cylinders with disjoint interiors. Problems differ according to whether the cylinders are finite or infinite and whether the aim is to maximize packing density or to prescribe contacts.
Bezdek and Kuperberg (1990) proved that the maximum packing density of congruent infinite right circular cylinders in three-dimensional Euclidean space is
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This bound is attained by parallel cylinders whose axes meet a perpendicular plane in a hexagonal lattice with nearest-neighbor spacing equal to the cylinder diameter.
Gardner's cigarette problem concerns congruent finite right circular cylinders that touch pairwise. Seven can be arranged in this way. The analogous infinite-cylinder question is Littlewood's cylinder problem, which is equivalent to finding seven equidistant lines in three-dimensional Euclidean space. Gardner's construction uses contacts at the end rims, and some of its axes intersect. Extending the cylinders infinitely in both directions would therefore cause their interiors to overlap, so this construction does not solve Littlewood's problem.
Kuperberg's six-cylinder conjecture instead concerns infinite right circular cylinders of radius 1 that have pairwise disjoint interiors and all touch a common unit ball, without requiring pairwise contact between the cylinders.
