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
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
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
The blue-edged piece is
concave outward; during the sweep (right to left)
its intersecton with
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."