We report the finding of a bowl-shaped quasi-planar structure of a B84 cluster with four hexagonal holes and a three-chain ring all around the edges using ab initio calculations. A large number of other isomers including those explored earlier such as an empty cage, a filled cage, and a disordered structure, have been found to lie in a significantly higher energy band. A tubular structure, however, is only about 0.45 eV higher in energy. Calculations of the infrared and Raman spectra show that the quasi-planar structure is dynamically stable. These results suggest that quasi-planar structures may be among the low energy structures for larger clusters as well. Accordingly we have calculated the optimal quasi-planar structures stabilized with 2, 3, 5, 6, and 7 hexagonal holes also. The stability of quasi-planar structures is discussed in terms of multi-center two-electron bonding and it is shown that with increasing size their binding energy tends to approach the value for an α-boron sheet.