Topology and Verb Classes
Feature Column Archive
4. The fold: ``The day begins.''
The fold catastrophe is exemplified by the critical point of the function f(x) = x3. This critical point is unstable: perturbing the function to f(x) = x3 + ax gives a function with a maximum and a minimum near zero if a is negative, no matter how small, and no critical point at all if a is positive. A typical one-dimensional section will follow the parameter a : the energy minumum that exists for a < 0 disappears as the section crosses a = 0 . The corresponding verb class is represented by ``The day ends.'' if the section is traversed from negative a to positive and ``The day begins.'' if it is traversed in the opposite direction.