Feature Column

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.''


Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Read more . . .

Search Feature Column

Feature Column at a glance


Show Archive

Browse subjects



Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia