Editing Soap bubble (section)

== Mathematics ==
Soap bubbles are physical examples of the complex [[math]]ematical problem of [[minimal surface]]. They will assume the shape of least [[surface area]] possible containing a given volume. A true minimal surface is more properly illustrated by a [[soap film]], which has equal pressure on both sides, becoming a surface with zero [[mean curvature]]. A soap bubble is a closed soap film: due to the difference in outside and inside pressure, it is a surface of ''constant'' mean curvature.

While it has been known since 1884 that a spherical soap bubble is the least-area way of enclosing a given volume of air (a theorem of [[H. A. Schwarz]]), it was not until 2000 that it was proven that two merged soap bubbles provide the optimum way of enclosing two given volumes of air of different size with the least surface area. This has been dubbed the ''[[double bubble conjecture]]''.<ref>{{ cite journal| last1 = Hutchings | first1 = Michael | first2= Frank |last2 = Morgan| first3=Manuel |last3= Ritoré | first4= Antonio| last4= Ros | date =  July 17, 2000 |  title = Proof of the double bubble conjecture | journal =   Electronic Research Announcements of the American Mathematical Society| volume = 6| issue = 6 |pages = 45–49 | doi = 10.1090/S1079-6762-00-00079-2
| doi-access = free | hdl = 10481/32449 | hdl-access = free }}</ref>

Because of these qualities, soap bubble films have been used in practical problem solving applications. [[Structural engineer]] [[Frei Otto]] used soap bubble films to determine the geometry of a sheet of least surface area that spreads between several points, and translated this geometry into revolutionary [[tension structure|tensile roof structures]].<ref>[https://www.theguardian.com/artanddesign/2004/oct/04/architecture Jonathan Glancey, The Guardian November 28, 2012] {{webarchive|url=https://web.archive.org/web/20170108193633/https://www.theguardian.com/artanddesign/2004/oct/04/architecture |date=January 8, 2017 }}</ref> A famous example is his West German Pavilion at Expo 67 in Montreal.

=== Soap bubbles as unconventional computing ===
{{see also|Unconventional computing}}
The structures that soap films make can not just be enclosed as spheres, but virtually any shape, for example in wire frames. Therefore, many different minimal surfaces can be designed. It is actually sometimes easier to physically make them than to compute them by [[mathematical modelling]]. This is why the soap films can be considered as [[analog computer]]s which can outperform conventional computers, depending on the complexity of the system.<ref>{{Cite journal | doi = 10.1511/2012.96.1| title = The Soap Film: An Analogue Computer| journal = American Scientist| volume = 100| issue = 3| pages = 1| year = 2012| last1 = Isenberg | first1 = Cyril }}</ref><ref>{{Cite journal | doi = 10.1511/2012.96.1| title = The Soap Film: An Analogue Computer| journal = American Scientist| volume = 64| issue = 3| pages = 514–518| year = 1976| last1 = Isenberg | first1 = Cyril | bibcode = 1976AmSci..64..514I}}</ref><ref>{{Cite journal | doi = 10.1511/2012.96.1| title = Soap Film Letters| journal = American Scientist| volume = 100| issue = January–February| pages = 1| year = 1977| last1 = Taylor | first1 = Jean E. }}</ref>