The structures and energies of isolobal (CH)n and (BCO)n polyhedral species, computed at the B3LYP density functional theory level, reveal contrasts in behavior. The strain energies of the (BCO)n cages are much smaller. Also unlike the (CH)n cages, the most stable (BCO)n polyhedra (n > or = 10) prefer structures with the largest number of three-membered rings. The planar (or nearly planar) faces of the cage systems were modeled by computations on planar, isoelectronic (CH2)n (Dnh) and (HBCO)n (Cnv) rings. While the strain energies of all the planar carbon rings, relative to the most stable D5h (CH2)5, were large, the strain energies of all the planar (HBCO)n (Cnv) rings were small. Remarkably, the three-membered (HBCO)3 (C3v) ring was the most stable. Finally, large (BCO)n systems prefer tubelike rather than cage structures.