# Topology and Verb Classes

## 7. Derivatives of the swallowtail: "She repulses his advances," etc.

The swallowtail catastrophe has two derivative catastrophes, one of which is syntactically interesting. If we slide a two-dimensional piece of surface across one of the cusp lines of the locus, the point of tangency will spawn one or the other of the catastrophes that Thom calls the lips and beak-to-beak, according as the tangency is on the outside (the surface curls into the blue region) or on the inside (the surface curls into the pink region). The additional parameter corresponding to the sliding across the cusp line stabilizes these catastrophes: small perturbations of the motion will not affect the topology.

 The beak-to-beak and lips catastrophes are exhibited at the point of tangency of curved pieces of surface with the cusp locus of the swallowtail. Embedding them in a one-parameter family (the surface sweeps across the locus) stabilizes them. The green-edged piece is concave inward; at the beginning of the sweep (left to right) it is entirely in the pink (two-minimum) region. Then its intersection with the pink region narrows to one central stripe, then two touching beaks, at the point of tangency, and then two facing beaks. The blue-edged piece is concave outward; during the sweep (right to left) its intersecton with the pink region goes from empty to one point (at tangency) to the lips configuration.

The beak to beak catastrophe has two one-dimensional sections with syntactic interpretations.

 The red section leads from two minima to one minimum and back to two minima. The verb class corresponding to this morphology is reflection or repulsion: "She repulses his advances."

 The yellow section also leads from two minima to one minimum and back to two minima; in this case the relative position of the two minima is interchanged. The verb class corresponding to this morphology is refraction or traversal: "He crosses the river."