Magic Gems: A Polyhedral Framework for Magic Squares

📝 Original Info

• Title: Magic Gems: A Polyhedral Framework for Magic Squares
• ArXiv ID: 2512.09170
• Date: 2025-12-09
• Authors: Kyle Elliott Mathewson

📝 Abstract

We introduce Magic Gems, a geometric representation of magic squares as threedimensional polyhedra. By mapping an n × 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. Building on prior work connecting magic squares to physical properties such as moment of inertia, this construction reveals an explicit statistical structure: we show that magic squares have vanishing covariances between position and value. We develop a covariance energy functional-the sum of squared covariances with individual row, column, and diagonal indicator variables-and prove that for all n ≥ 3, an arrangement is a magic square if and only if this complete energy vanishes. This characterization transforms the classical line-sum definition into a statistical orthogonality condition. We also study a simpler "low-mode" relaxation using only four aggregate position indicators; this coincides with the complete characterization for n = 3 (verified exhaustively) but defines a strictly larger class for n ≥ 4 (explicit counterexamples computed). Perturbation analysis demonstrates that magic squares are isolated local minima in the energy landscape. The representation is invariant under dihedral symmetry D 4 , yielding canonical geometric objects for equivalence classes.

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