We present the results of analytical calculations and numerical simulations of the behavior of a new class of chain molecules that we call thick polymers. The concept of the thickness of such a polymer, viewed as a tube, is encapsulated by a special three-body interaction and impacts on the behavior both locally and nonlocally. When thick polymers undergo compaction because of an attractive self-interaction, we find a new type of phase transition between a compact phase and a swollen phase at zero temperature with increasing thickness. In the vicinity of this transition, short tubes form space-filling helices and sheets as observed in protein native-state structures. With increasing chain length, or with an increasing number of chains, we numerically find a crossover from secondary-structure motifs to a quite distinct class of structures akin to the semicrystalline phases of polymers or amyloid fibers in polypeptides. (C) 2005 Wiley Periodicals, Inc.