''The Swallow's Tail — Series of Catastrophes'' (french: La queue d'aronde — Série des catastrophes) was

Salvador Dalí Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (; ; ; 11 May 190423 January 1989) was a Spanish surrealist artist renowned for his technical skill, precise draftsmanship, and the striking and bizarre images in ...

's last painting. It was completed in May 1983, as the final part of a series based on the mathematical catastrophe theory of René Thom. Thom suggested that in four-dimensional phenomena, there are seven possible equilibrium surfaces, and therefore seven possible discontinuities, or "elementary catastrophes": fold,

cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...

, swallowtail,

butterfly Butterflies are insects in the macrolepidopteran clade Rhopalocera from the order Lepidoptera, which also includes moths. Adult butterflies have large, often brightly coloured wings, and conspicuous, fluttering flight. The group comprise ...

, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic. "The shape of Dalí'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 The cello ( ; plural ''celli'' or ''cellos'') or violoncello ( ; ) is a bowed (sometimes plucked and occasionally hit) string instrument of the violin family. Its four strings are usually tuned in perfect fifths: from low to high, C2, G ...

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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

: ∫." In his 1979 speech, ''Gala, Velázquez and the Golden Fleece'', presented upon his 1979 induction into the prestigious

Académie des Beaux-Arts An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosophy, ...

of the

Institut de France The (; ) is a French learned society, grouping five , including the Académie Française. It was established in 1795 at the direction of the National Convention. Located on the Quai de Conti in the 6th arrondissement of Paris, the institute ...

, Dalí described Thom's theory of catastrophes as "the most beautiful aesthetic theory in the world".Dalí, Salvador, 'Gala, Velásquez and the Golden Fleece' (9 May 1979). Reproduced in-part in: Originally published in French as ''Dalí, l'oeuvre et l'homme'' (Lausanne: Edita, 1984). He also recollected his first and only meeting with René Thom, at which Thom purportedly told Dalí that he was studying

tectonic plate Plate tectonics (from the la, label=Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is the generally accepted scientific theory that considers the Earth's lithosphere to comprise a number of large te ...

s; this provoked Dalí 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. Thom reportedly replied, "I can assure you that Spain pivoted precisely — not in the area of — but exactly there where the Railway Station in Perpignan stands today". Dalí was immediately enraptured by Thom's statement, influencing his painting ''Topological Abduction of Europe — Homage to René Thom'', the lower left corner of which features an equation closely linked to the "swallow's tail":

V = x^5 + ax^3 + bx^2 + cx,

an illustration of the graph, and the term ''queue d'aronde''. The seismic fracture that transverses ''Topological Abduction of Europe'' reappears in ''The Swallow's Tail'' at the precise point where the y-axis of the swallow's tail graph intersects with the S-curve of the

References

{{DEFAULTSORT:Swallow's Tail, The Paintings by Salvador Dalí 1983 paintings Mathematics and culture Singularity theory Mathematical artworks Musical instruments in art Collection of the Dalí Theatre and Museum