Topology and Verb ClassesFeature 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) = x^{5} at x = 0 . This critical point is unstable: the most general perturbation requires three parameters: f(x) = x^{5} + ax^{3} + bx^{2} + cx. The possible numbers of minima for a degreefive curve are zero, one and two, shown as the greensided, bluesided and pinksided regions in this figure:
The catastrophe locus in the space of parametersa, b, c for the most general perturbation of f(x) = x^{5}.The locus separates perturbations with no minimum (greensided region),perturbations with one minumum (bluesided region) and perturbationswith two minima (pinksided region). 
The red line is a onedimensional 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.''  

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