The lost theorem (Q1272410)

While working on a problem on magic squares posed by Martin Gardner, the author devised a correspondence between parallelograms drawn on the plane and equivalence classes of eight complex \(3\times 3\) magic squares. This correspondence explains the presence of the eight 3-term arithmetic progressions that can be formed from the nine numbers in a magic square in light of the geometry of the parallelogram. The author also discusses the Chinese magic square known as \textit{Lo shu} that dates back at least to the fourth century BC, and finds it not quite as aesthetically pleasing as the symmetry of the magic square formed from the Gaussian integers in the set \(\{0, \pm 1, \pm{\mathbf i}, \pm 1\pm{\mathbf i}\}\).