  Chain molecules play a key role in the polymer field and in living cells. Our
focus is on a new homopolymer model of a linear chain molecule subject to an
attractive self-interaction promoting compactness. We analyze the model using
simple analytic arguments complemented by extensive computer simulations. We
find several striking results: there is a first order transition from a high
temperature random coil phase to a highly unusual low temperature phase; the
modular ground states exhibit significant degeneracy; the ground state
structures exhibit spontaneous dimensional reduction and have a two-layer
structure; and the ground states are assembled from secondary motifs of helices
and strands connected by tight loops. We discuss the similarities and notable
differences between the ground state structures (we call these PoSSuM -- Planar
Structures with Secondary Motifs) in the novel phase and protein native state
structures.
