Topology and Verb Classes

6. The swallowtail: "The bird lays an egg, swallows it and dies."

The swallowtail catastrophe is exemplified by the inflection point of the function f(x) = x⁵ at x = 0 . This critical point is unstable: the most general perturbation requires three parameters: f(x) = x⁵ + ax³ + bx² + cx. The possible numbers of minima for a degree-five curve are zero, one and two, shown as the green-sided, blue-sided and pink-sided regions in this figure:

The catastrophe locus in the space of parameters a, b, c for the most general perturbation of f(x) = x⁵. The locus separates perturbations with no minimum (green-sided region), perturbations with one minumum (blue-sided region) and perturbations with two minima (pink-sided region).

The red line is a one-dimensional section of parameter space which, read going upwards, has a process change from one minimum to two minima, then back to one minimum which disappears. The corresponding verb class is suicide: the subject emits an instrument which then kills it: "The bird lays an egg, swallows it, and dies."