This paper introduces a novel approach for reassembling pot sherds found at archaeological excavation sites, for the purpose of reconstructing clay pots that had been made on a wheel. These pots and the sherds into which they have broken are axially symmetric. The reassembly process can be viewed as 3D puzzle solving or generalized cylinder learning from broken fragments. The estimation exploits both local and semi-global geometric structure, thus making it a fundamental problem of geometry estimation from noisy fragments in computer vision and pattern recognition. The data used are densely digitized 3D laser scans of each fragment's outer surface. The proposed reassembly system is automatic and functions when the pile of available fragments is from one or multiple pots, and even when pieces are missing from any pot. The geometric structure used are curves on the pot along which the surface had broken and the silhouette of a pot with respect to an axis, called axis-profile curve (APC). For reassembling multiple pots with or without missing pieces, our algorithm estimates the APC from each fragment, then reassembles into configurations the ones having distinctive APC. Further growth of configurations is based on adding remaining fragments such that their APC and break curves are consistent with those of a configuration. The method is novel, more robust and handles the largest numbers of fragments to date.