Define gratuitous symmetry-breaking

Numerical-optimization.md

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 ### Gratuitous symmetry-breaking
 
-_To be added_
+In some cases, a configuration has a symmetry that must be broken to satisfy a newly imposed constraint. This happens, for example, when a point at the origin is constrained to lie on a sphere centered at the origin. In other cases, however, a configuration has a symmetry which is logically and aesthetically compatible with a newly imposed constraint. This happens, for example, when three equal-curvature spheres that intersect each other at 90° angles are constrained to instead intersect each other at 60° angles. A user might reasonably expect the realization process to preserve this symmetry, at least approximately. When it doesn’t, we’ll say the symmetry has been _gratuitously broken_.
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+The idea of gratuitous symmetry-breaking can be extended to approximate symmetries. For example, suppose three spheres of roughly the same curvature intersect each other at roughly 90° angles. If we constrain these spheres to intersect each other at 60° angles, we might reasonably expect their curvatures to stay roughly the same as each other. If the curvatures instead come out wildly different from each other, we can say that the approximate symmetry of the initial configuration has been gratuitously broken.
 
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