Hassler Whitney proved a result that explains this – any smooth map from the plane to the plane can be perturbed so that all its singularities are folds and cusps; in particular, the singularities of "generic" such maps can only be folds and cusps: https://www.jstor.org/stable/1970070 2/5
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This is one of the classic results of "singularity theory", which studies how spaces and functions go bad (they always seem to, don't they?). Here's an animation of a more exotic singularity called a "dove" from Herwig Hauser's gallery of surfaces: https://homepage.univie.ac.at/herwig.hauser/gallery.html … 3/5pic.twitter.com/E8MSUi6r4m
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Singularity theory gave birth to "catastrophe theory", one of the great mathematical fads of the 70's. Here's an image from a Scientific American article by Zeeman http://www.gaianxaos.com/pdf/dynamics/zeeman-catastrophe_theory.pdf … illustrating a model for aggression in dogs which exhibits folds meeting in a cusp!
3/5pic.twitter.com/3WygSb6QwK
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Catastrophe theory was applied broadly: to bubbles and crashes in economics, military action and war in history and even to prison riots! The backlash was harsh: see these wise words on mathematical modeling from Smale's rather devastating book review: https://projecteuclid.org/download/pdf_1/euclid.bams/1183541477 … 4/5pic.twitter.com/tZE3gZSS2p
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Though, I can't blame the catastrophe theorists for getting excited; the underlying math and geometry is really pretty! For a very clear and sober introduction to catastrophes, see the article "Dangerous intersection" by
@stevenstrogatz: https://opinionator.blogs.nytimes.com/2012/10/08/dangerous-intersection/ … 5/5Show this thread
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