A typical protein structure is a compact packing of connected alpha-helices and/or beta-strands. We have developed a method for generating the ensemble of compact structures a given set of helices and strands can form. The method is tested on structures composed of four alpha-helices connected by short turns. All such natural four-helix bundles that are connected by short turns seen in nature are reproduced to closer than 3.6 A per residue within the ensemble. Because structures with no natural counterpart may be targets for ab initio structure design, the designability of each structure in the ensemble-defined as the number of sequences with that structure as their lowest-energy state-is evaluated using a hydrophobic energy. For the case of four alpha-helices, a small set of highly designable structures emerges, most of which have an analog among the known four-helix fold families; however, several packings and topologies with no analogs in protein database are identified.