Magic Square ↔ Electron Orbital Mapping

The 16 cells of a magic square can be mapped to electron configurations. Each cell's value determines its shell distance, while its grid position determines its angular orientation.

Shell Configuration (by distance)

Orbital Notation (Atomic Analogy)

Mapping to Real Orbitals

Magic Square	Atomic Analog
Shell distance = value	Principal quantum number (n)
Grid position angle	Magnetic quantum number (ml)
Row/Col index	Angular momentum (l)
Covariance = 0	Energy eigenstate
880 magic squares	Allowed electron configurations

Balance as Quantum Constraint

Just as electrons fill orbitals following the Aufbau principle and Pauli exclusion, magic square values fill grid positions following the covariance constraints.

The 6 covariance conditions act like quantum rules:

• c₁, c₂ = 0 → Center of mass at nucleus (like s-orbital symmetry)
• c₃, c₄ = 0 → Moment balanced (like d-orbital shapes)
• c₅, c₆ = 0 → Diagonal slices balanced (like nodal planes)

Shell Occupancy Pattern

Comparison with Noble Gas Configurations

Helium (2): 1s² → 2 particles in innermost shell
Neon (10): 1s² 2s² 2p⁶ → filled shells
Argon (18): [Ne] 3s² 3p⁶

Magic Square (16): A unique "element" with 16 particles distributed across shells 1-16, constrained to perfect balance.