We report the results of first-principles calculations showing that boron can form a wide variety of metastable planar and tubular forms with unusual electronic and mechanical properties. The preferred planar structure is a buckled triangular lattice that breaks the threefold ground state degeneracy of the flat triangular plane. When the plane is rolled into a tube, the ground state degeneracy leads to a strong chirality dependence of the binding energy and elastic response, an unusual property that is not found in carbon nanotubes. The achiral (n, 0) tubes derive their structure from the flat triangular plane. The achiral (n, n) boron nanotubes arise from the buckled plane, and have large cohesive energies and novel structures as a result.