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How does three-centre bonding help boranes form cage-like structures?

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A labeled diagram of the periodic table entry for boron shows its atomic number 5, symbol B, electron configuration, atomic weight range, and classification as a solid, weakly acidic, other nonmetal with a rhombohedral crystal structure at room temperature. — boron Properties of boron.

Boranes are electron-deficient compounds that utilize multicentre bonding to achieve stability, a feature that distinguishes them from the simple chain or ring configurations found in carbon chemistry.12

The formation of cage-like structures is driven by the ability of boron atoms to participate in three-centre, two-electron bonds (3c,2e bonds). In these bonds, a single pair of electrons is delocalized over three atoms rather than the traditional two, allowing boron to form intricate, polyhedral networks.12

This bonding mechanism facilitates the creation of various structural frameworks:
* Closo-boranes: Closed, cage-like polyhedrons.12
* Nido-boranes: Nonclosed structures resembling birds' nests, where the cluster occupies corners of a polyhedron with one missing vertex.12
* Arachno-boranes: More open, spiderweb-like structures with two missing vertices.12

Chemists use Wade’s rules to correlate the number of valence electrons with the specific polyhedral geometry of these clusters, rationalizing how the combination of 3c,2e and ordinary 2c,2e bonds builds these complex, stable architectures.1