Extensive optimisation calculations are performed for the B80 isomers in order to find out which principles underlie the formation of large hollow boron cages. Our analysis shows that the most stable isomers contain triangular B10 or rhombohedral B16 building blocks. The lowest-energy isomer has C3v symmetry and is characterised by a belt of three interconnected B16 units and two separate B10 units. At the B3LYP/6-31G(d) level of theory, this newly discovered isomer is 2.29, 1.48, and 0.54 eV below the leapfrog B80 of Szwacki et al., the Th-B80 of Wang, and the D3d-B80 of Pochet et al., respectively. Our C3v isomer is therefore identified as the most stable hollow cage isomer of B80 presently known. Its HOMO–LUMO gap of 1.6 eV approaches that of the leapfrog B80. The leapfrog principle still remains a reliable scheme for producing boron cages with larger HOMO–LUMO gaps, whereas the thermodynamically most stable B80 cages are formed when all pentagonal faces are capped. We show that large hollow cages of boron retain a preference for fullerene frames. The additional capping is in accordance with the following rules: preference for capping of pentagonal faces, formation of B10 and/or B16 units, homogeneous distribution of the hexagonal caps, and hole density approaching 1/9. Although our most stable B80 isomer still remains higher in energy than the B80 core–shell structure, we show that by applying the bonding principles to larger structures it is possible to construct boron cages with higher stabilisation energy per boron atom than the core–shell structure; a prototypical example is B160. This clearly shows the continuous competition between the two suggested construction schemes, namely, the formation of multiple-shell structures and hollow cages.