saddles

A saddle in mathematics is a shape or point on a surface where the curvature is mixed—curving upward in one direction and downward in another, like a horse saddle. In optimization, a saddle point is a position that is a minimum along one direction and a maximum along another, representing a critical point that’s not purely a high or low point. Think of a mountain pass: it’s higher than surrounding valleys but lower than peaks on either side. Saddles are important in understanding the structure of complex systems, helping identify optimal solutions and stability.