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