From f699015a3d1d5ba05a1ca2ac16fdfcf840c2a3fd Mon Sep 17 00:00:00 2001 From: Vectornaut Date: Thu, 6 Nov 2025 08:46:06 +0000 Subject: [PATCH] Define gratuitous symmetry-breaking --- Numerical-optimization.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/Numerical-optimization.md b/Numerical-optimization.md index 6665b34..e1f75ef 100644 --- a/Numerical-optimization.md +++ b/Numerical-optimization.md @@ -29,7 +29,9 @@ _To be added_ ### 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_. + +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. ## Methods