Magic Gems: A Polyhedral Framework for Magic Squares

We introduce Magic Gems, a geometric representation of magic squares as three-dimensional polyhedra. By mapping an n x n magic square onto a centered coordinate grid with cell values as vertical displacements, we construct a point cloud whose convex hull defines the Magic Gem. This reveals a connection between magic square constraints and statistical structure: we prove that magic squares have vanishing covariances between position and value. We introduce a covariance energy functional -- the sum of squared covariances with row, column, and diagonal indicator variables -- and prove for n=3 (via exhaustive enumeration) that its zeros are precisely the magic squares. Large-scale sampling for n=4,5 (460+ million arrangements) provides strong numerical evidence that this characterization extends to larger orders. Perturbation analysis demonstrates that magic squares are isolated local minima. The representation is invariant under dihedral symmetry D_4, yielding canonical geometric objects for equivalence classes.