Single covalent bonds are formed when two atoms share an edge, as in structure C below. This results in the sharing of two electrons. Ionic bonds are formed by the transfer of an electron from one cube to another without sharing an edge (structure A). An intermediate state where only one corner is shared (structure B) was also postulated by Lewis.
Double bonds are formed by sharing a face between two cubic atoms. This results in sharing four electrons:
Triple bonds could not be accounted for by the cubical atom model, because there is no way of having two cubes share three parallel edges. Lewis suggested that the electron pairs in atomic bonds have a special attraction, which result in a tetrahedral structure, as in the figure below (the new location of the electrons is represented by the dotted circles in the middle of the thick edges). This allows the formation of a single bond by sharing a corner, a double bond by sharing an edge, and a triple bond by sharing a face. It also accounts for the free rotation around single bonds and for the tetrahedral geometry of methane.
Lewis, Gilbert N. (1916-04-01). "The Atom and the Molecule". Journal of the American Chemical Society. 38 (4): 762–785. doi:10.1021/ja02261a002. S2CID 95865413. http://scarc.library.oregonstate.edu/coll/pauling/bond/papers/corr216.3-lewispub-19160400.html ↩
Langmuir, Irving (1919-06-01). "The Arrangement of Electrons in Atoms and Molecules". Journal of the American Chemical Society. 41 (6): 868–934. doi:10.1021/ja02227a002. https://zenodo.org/record/1429026 ↩