  We develop a simple analytical theory that relates dense sphere packings in a
cylinder to corresponding disk packings on its surface. It applies for ratios
R=D/d (where d and D are the diameters of the hard spheres and the bounding
cylinder, respectively) up to R=1+1/sin(pi/5). Within this range the densest
packings are such that all spheres are in contact with the cylindrical
boundary. The detailed results elucidate extensive numerical simulations by
ourselves and others by identifying the nature of all competing phases.
