Iron filings / line fields
Line charges create a field over the square
Rows, columns, and diagonals act like charged field lines. Animated filings align with the local residual field. A magic square makes the field collapse toward neutrality.
Whole-system magnetic flow
Charged values flow around one another
Values are continuous bubbles, not swapped pairs. Every bubble feels all row, column, and diagonal fields at once, repels nearby bubbles, and is softly pulled toward lattice wells. The displayed square is the nearest-cell assignment induced by the flow.
Spin lattice
Cells as frustrated spins
Each cell’s spin is driven by its centered height and by the line residuals passing through it. Large coherent regions reveal imbalanced rows, columns, or diagonals.
Planets / electron orbitals
Values orbit on shells from their vector lengths
The original vector (X,Y,Z) sets each body’s shell
radius. Color and glow are driven by each cell’s line-force
frustration, tying the orbit picture back to magic energy.
Simulated annealing swaps
Values snap toward lower line energy
A proposed swap is accepted if it reduces energy, or with
probability exp(-ΔE/T). The chart tracks how a random
permutation cools toward balance.
Energy landscape
Nearby permutations as terrain
Points are sampled one-swap and random-walk neighbors. Axes are low-mode energy and high-mode residual energy; color is complete line energy.