A compendium of unique haeckelite boron and aluminum-group V binary materials have been assessed for their fundamental thermodynamic and ground state electronic properties within density functional theory. We explore their thermodynamic stability relative to new bulk haeckelite crystal structures and find a number of stable polymorphs of planar and buckled ultrathin nanosheets. The bulk boron and aluminum haeckelite crystals display semiconducting and metallic behavior. From the dispersion curves, we predict the formation of both indirect and direct bandgap crystals. We also discover the existence of a five-coordinate aluminum antimonide crystal hitherto never before observed. Moreover, it is found that a number of the Archimedean four and eight membered ring tessellation planar nanosheets could form should synthesis be attempted. It is predicted that these nanosheets can attain two configurations - planar and buckled. From this work we find that combinations of elements such as boron and nitrogen or phosphorus, and aluminum and nitrogen will likely become true single-atom thick nanosheets. These materials show intrinsic indirect bandgap character, which spans the ultraviolet, visible, and infrared spectrum. In the boron series of these materials, the planar structures show double extrema in the bandstructures with van Hove singularities in the projected density of states at the Fermi energy suggesting strong light-matter interactions. The aluminum series we observe strong charge transfer and larger indirect bandgap nanosheets. This study serves as a starting point for a new class of inorganic bulk and ultrathin film materials, which can have many varied applications in nanotechnology.