{{Short description|Mathematical surface of constant unit negative Gaussian curvature}} The '''Kuen surface''' is a mathematical surface of constant negative unit Gaussian curvature, making it an example of a pseudospherical surface.<ref>{{Cite web |title=Kuen Surface |url=https://virtualmathmuseum.org/Surface/kuen/kuen.html |access-date=2025-03-28 |website=virtualmathmuseum.org}}</ref><ref name=VMM>{{Cite web |title=Three Pseudospherical Surfaces, Dini Family, Kuen, Breather |url=https://virtualmathmuseum.org/docs/DiniKuenBreather.pdf |access-date=29 March 2025 |website=virtualmathmuseum.org}}</ref> It can be described as a parametric surface<ref name=VMM/> in terms of the parametric equations
:<math>x = 2 \cosh v \, (\cos u + u \sin u) / w</math> :<math>y = 2 \cosh v \, ( \sin u - u \cos u ) / w</math> :<math>z = v - (2 \sinh v \cosh v ) / w</math>
where
:<math>w = (\cosh v)^2 + u^2</math>
It is named after, and was first described by, the German mathematician Theodor Kuen in 1884.<ref>{{Cite web |title=Kuen surface |url=https://www.mathcurve.com/surfaces.gb/kuen/kuen.shtml |access-date=2025-03-28 |website=www.mathcurve.com}}</ref><ref>Kuen, T. "Ueber Flächen von constantem Krümmungsmass." Sitzungsber. d. königl. Bayer. Akad. Wiss. Math.-phys. Classe, Heft II, 193-206, 1884.</ref> The surface is a special case of the class of Enneper surfaces, first described by Alfred Enneper.
The Kuen surface was of interest to surrealist artists, including Max Ernst and Man Ray.<ref>{{Cite web |title=Max Ernst - The historical mathematical models of Mathematics department |url=https://mostre.cab.unipd.it/modellimatematici/en/59/max-ernst |access-date=2025-03-28 |website=mostre.cab.unipd.it}}</ref> The surface has also inspired work by the Japanese sculptor Toshimasa Kikuchi.<ref>{{Cite web |date=2021-08-31 |title=Toshimasa Kikuchi's Slender Sculptures |url=https://pen-online.com/arts/toshimasa-kikuchis-slender-sculptures/?scrolled=0 |access-date=2025-03-28 |website=Pen Magazine International |language=en}}</ref>
== References == {{reflist}}
== See also == * Dini's surface * Breather surface
== External links == * [https://americanhistory.si.edu/collections/object/nmah_693971 A plaster model of the Kuen surface] in the collection of the National Museum of American History * [https://paulbourke.net/geometry/kuen/ 3D renderings of the Kuen surface] by Paul Bourke
Category:Algebraic surfaces Category:Surfaces of constant curvature
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