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.