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Chain molecules play a key role in the polymer field and in living cells. Ourfocus is on a new homopolymer model of a linear chain molecule subject to anattractive self-interaction promoting compactness. We analyze the model usingsimple analytic arguments complemented by extensive computer simulations. Wefind several striking results: there is a first order transition from a hightemperature random coil phase to a highly unusual low temperature phase; themodular ground states exhibit significant degeneracy; the ground statestructures exhibit spontaneous dimensional reduction and have a two-layerstructure; and the ground states are assembled from secondary motifs of helicesand strands connected by tight loops. We discuss the similarities and notabledifferences between the ground state structures (we call these PoSSuM -- PlanarStructures with Secondary Motifs) in the novel phase and protein native statestructures.

Spontaneous dimensional reduction and novel ground state degeneracy in a simple chain model