The Kuen surface is a mathematical surface of constant negative unit Gaussian curvature, making it an example of a pseudospherical surface.^[1][2] It can be described as a parametric surface^[2] in terms of the parametric equations

${\displaystyle x=2\cosh v\,(\cos u+u\sin u)/w}$

${\displaystyle y=2\cosh v\,(\sin u-u\cos u)/w}$

${\displaystyle z=v-(2\sinh v\cosh v)/w}$

where

${\displaystyle w=(\cosh v)^{2}+u^{2}}$

It is named after, and was first described by, the German mathematician Theodor Kuen in 1884.^[3][4] 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.^[5] The surface has also inspired work by the Japanese sculptor Toshimasa Kikuchi.^[6]

• Dini's surface
• Breather surface

• A plaster model of the Kuen surface in the collection of the National Museum of American History
• 3D renderings of the Kuen surface by Paul Bourke

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