The Swallow’s Tail - Series on Catastrophes, 1983 by Salvador Dali
The Swallow’s Tail - Series of Catastrophes (French: La queue d’aronde - Serie des catastrophes) was Salvador Dali’s last painting. It was completed in May 1983, as the final part of a series based on the mathematical catastrophe theory of Rene Thom.
Thom suggested that in four-dimensional phenomena, there are seven possible equilibrium surfaces, and therefore seven possible discontinuities, or “elementary catastrophes”: fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic. The shape of Dali’s Swallow’s Tail is taken directly from Thom’s four-dimensional graph of the same title, combined with a second catastrophe graph, the s-curve that Thom dubbed, ‘the cusp’. Thom’s model is presented alongside the elegant curves of a cello and the instrument’s f-holes, which, especially as they lack the small pointed side-cuts of a traditional f-hole, equally connote the mathematical symbol for an integral in calculus.
In his 1979 speech, “Gala, Velazquez and the Golden Fleece”, presented upon his 1979 induction into the prestigious Academie des Beaux-Arts of the Institut de France, Dali described Thom’s theory of catastrophes as “the most beautiful aesthetic theory in the world”. He also recollected his first and only meeting with Rene Thom, at which Thom purportedly told Dali that he was studying tectonic plates; this provoked Dali to question Thom about the railway station at Perpignan, France (near the Spanish border), which the artist had declared in the 1960s to be the center of the universe.
https://www.dalipaintings.com/the-swallows-tail-series-on-catastrophes.jsp
A simple example of the behaviour studied by catastrophe theory is the change in shape of an arched bridge as the load on it is gradually increased. The bridge deforms in a relatively uniform manner until the load reaches a critical value, at which point the shape of the bridge changes suddenly—it collapses. While the term catastrophe suggests just such a dramatic event, many of the discontinuous changes of state so labeled are not. The reflection or refraction of light by or through moving water is fruitfully studied by the methods of catastrophe theory, as are numerous other optical phenomena. More speculatively, the ideas of catastrophe theory have been applied by social scientists to a variety of situations, such as the sudden eruption of mob violence.
https://www.britannica.com/science/catastrophe-theory-mathematics
René Frédéric Thom (2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as the founder of catastrophe theory (later developed by Christopher Zeeman).
https://en.m.wikipedia.org/wiki/René_Thom
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.
Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide.
Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s. It considers the special case where the long-run stable equilibrium can be identified as the minimum of a smooth, well-defined potential function (Lyapunov function). Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.
In the late 1970s, applications of catastrophe theory to areas outside its scope began to be criticized, especially in biology and social sciences. Zahler and Sussmann, in a 1977 article in Nature, referred to such applications as being “characterised by incorrect reasoning, far-fetched assumptions, erroneous consequences, and exaggerated claims”. As a result, catastrophe theory has become less popular in applications.
Catastrophe theory analyzes degenerate critical points of the potential function — points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the germs of the catastrophe geometries. The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters.
When the degenerate points are not merely accidental, but are structurally stable, the degenerate points exist as organising centres for particular geometric structures of lower degeneracy, with critical features in the parameter space around them. If the potential function depends on two or fewer active variables, and four or fewer active parameters, then there are only seven generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed by diffeomorphism (a smooth transformation whose inverse is also smooth).[citation needed] These seven fundamental types are now presented, with the names that Thom gave them.
Catastrophe theory studies dynamical systems that describe the evolution of a state variable x over time t
In the above equation, V is referred to as the potential function, and u is often a vector or a scalar which parameterise the potential function. The value of u may change over time, and it can also be referred to as the control variable. In the following examples, parameters like a , b are such controls.
Swallowtail catastrophe
V = x5 + ax3 + bx2 + cx
The control parameter space is three-dimensional. The bifurcation set in parameter space is made up of three surfaces of fold bifurcations, which meet in two lines of cusp bifurcations, which in turn meet at a single swallowtail bifurcation point.
As the parameters go through the surface of fold bifurcations, one minimum and one maximum of the potential function disappear. At the cusp bifurcations, two minima and one maximum are replaced by one minimum; beyond them the fold bifurcations disappear. At the swallowtail point, two minima and two maxima all meet at a single value of x. For values of a > 0, beyond the swallowtail, there is either one maximum-minimum pair, or none at all, depending on the values of b and c. Two of the surfaces of fold bifurcations, and the two lines of cusp bifurcations where they meet for a < 0, therefore disappear at the swallowtail point, to be replaced with only a single surface of fold bifurcations remaining. Salvador Dalí’s last painting, The Swallow’s Tail, was based on this catastrophe.
https://en.m.wikipedia.org/wiki/Catastrophe_theory
