Topology and Verb Classes

Feature Column Archive

3. The simple minimum: ``I am."

The simple minimum is exemplified by the minimum of the function f(x) = x². This minimum is stable: every small perturbation of f(x) = x² will have a minumum near x=0. There is no possibility of a system with this internal dynamics jumping from one state to another: it is always at the single critical point. Any one-dimensional section through any parameter space will be constant, or nearly constant, near x=0. The corresponding verb class is represented by ``I am.'':

• 1. Natural languages, human thought and topology

   

• 2. Thom's axioms

   

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• 4. The fold: ``The day begins.''

   

• 5. The cusp: ``The fish swallows the bug, '' etc.

   

• 6. The swallowtail: ``The bird lays an egg, swallows it and dies.''

   

• 7. Derivatives of the swallowtail: ``She repulses his advances,'' etc.

   

• References