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Topology and Verb Classes

Feature Column Archive

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 parametersa, 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 perturbationswith 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.''