What is a guage?
I looked at the wiki article, and it's not clicking.
If someone could describe it through an illustration that would
be best.

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In electrodynamics, the underlying properties of the electric and magnetic fields are described by the scalar and vector potentials. While the fields are unique for a given problem, the potentials are not. There is a degree of freedom when choosing our potentials. This causes us to define additional constraints that allow us to find unique solutions to the potentials despite their invariance on the physical solution. This allows consistency in the results and can ease calculations. The shifting of the potentials within the degrees of freedom that still give rise to the same physical fields is a gauge transformation.

One set of constraints is called the Lorenz condition. The Lorenz condition does not fully constrain the potentials but it does give rise to a reduction in the mathematics. This is because it allows us to express the scalar and vector potentials as decoupled inhomogeneous wave equations. By decoupling the potentials, it allows us to use simpler mathematical methods to derive their solutions, like using a dyadic Green's function to relate the sources to the potentials.