Kuen surface

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]

References

1. "Kuen Surface". virtualmathmuseum.org. Retrieved 2025-03-28.
2. "Three Pseudospherical Surfaces, Dini Family, Kuen, Breather" (PDF). virtualmathmuseum.org. Retrieved 29 March 2025.
3. "Kuen surface". www.mathcurve.com. Retrieved 2025-03-28.
4. Kuen, T. "Ueber Flächen von constantem Krümmungsmass." Sitzungsber. d. königl. Bayer. Akad. Wiss. Math.-phys. Classe, Heft II, 193-206, 1884.
5. "Max Ernst - The historical mathematical models of Mathematics department". mostre.cab.unipd.it. Retrieved 2025-03-28.
6. "Toshimasa Kikuchi's Slender Sculptures". Pen Magazine International. 2021-08-31. Retrieved 2025-03-28.

See also

• Dini's surface
• Breather surface

External links

• 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