Contents 1 Laws for soap films 2 See also 3 Notes 4 Sources 5 External links


Laws for soap films[edit] Plateau's laws describe the shape and configuration of soap films as follows:[2] Soap films are made of entire (unbroken) smooth surfaces. The mean curvature of a portion of a soap film is everywhere constant on any point on the same piece of soap film. Soap films always meet in threes along an edge called a Plateau border, and they do so at an angle of arccos(−1/2) = 120°. These Plateau borders meet in fours at a vertex, and they do so at an angle of arccos(−1/3) ≈ 109.47° (the tetrahedral angle). Configurations other than those of Plateau's laws are unstable, and the film will quickly tend to rearrange itself to conform to these laws.[3] That these laws hold for minimal surfaces was proved mathematically by Jean Taylor using geometric measure theory.[4][5]


See also[edit] Young–Laplace equation, governing the curvature of surfaces in a soap film


Notes[edit] ^ Ball, 2009. pp. 66–71, 97–98, 291–292 ^ Ball, 2009. p. 68 ^ Ball, 2009. pp. 66–67 ^ Taylor, Jean E. (1976), "The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces", Annals of Mathematics, Second Series, 103 (3): 489–539, doi:10.2307/1970949, MR 0428181 . ^ Almgren, Frederick J., Jr.; Taylor, Jean E. (July 1976), "The geometry of soap films and soap bubbles", Scientific American, 235: 82–93, doi:10.1038/scientificamerican0776-82 .


Sources[edit] Ball, Philip (2009). Shapes. Nature's Patterns: a tapestry in three parts. Oxford University Press. pp. 66–71, 97–98, 291–292. ISBN 978-0-19-960486-9. 


External links[edit] Weisstein, Eric W. "Plateau's Laws". MathWorld.  v t e Patterns in nature Patterns Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry in crystals Quasicrystals in flowers in biology Tessellation Vortex street Wave Widmanstätten pattern Causes Pattern formation Biology Natural selection Camouflage Mimicry Sexual selection Mathematics Chaos theory Fractal Logarithmic spiral Physics Crystal Fluid dynamics Plateau's laws Self-organization People Plato Pythagoras Empedocles Leonardo Fibonacci Liber Abaci Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson On Growth and Form Alan Turing The Chemical Basis of Morphogenesis Aristid Lindenmayer Benoît Mandelbrot How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Related Pattern recognition Emergence Mathematics and art Retrieved from "https://en.wikipedia.org/w/index.php?title=Plateau%27s_laws&oldid=753777101" Categories: Minimal surfaces


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EnlargeTetrahedral AngleSoap FilmJoseph PlateauPatterns In NatureMean CurvatureTetrahedral AngleMinimal SurfaceJean TaylorGeometric Measure TheoryYoung–Laplace EquationJean TaylorAnnals Of MathematicsDigital Object IdentifierMathematical ReviewsFrederick J. Almgren, Jr.Jean TaylorScientific AmericanDigital Object IdentifierInternational Standard Book NumberSpecial:BookSources/978-0-19-960486-9Eric W. WeissteinMathWorldTemplate:Patterns In NaturePatterns In NatureFractureDuneFoamMeanderPhyllotaxisSoap BubbleSymmetryCrystal SymmetryQuasicrystalsFloral SymmetrySymmetry In BiologyTessellationVortex StreetWaveWidmanstätten PatternPattern FormationBiologyNatural SelectionCamouflageMimicrySexual SelectionMathematicsChaos TheoryFractalLogarithmic SpiralPhysicsCrystalFluid DynamicsSelf-organizationPlatoPythagorasEmpedoclesLeonardo FibonacciLiber AbaciAdolf ZeisingErnst HaeckelJoseph PlateauWilson BentleyD'Arcy Wentworth ThompsonOn Growth And FormAlan TuringThe Chemical Basis Of MorphogenesisAristid LindenmayerBenoit MandelbrotHow Long Is The Coast Of Britain? Statistical Self-Similarity And Fractional DimensionPattern Recognition (psychology)EmergenceMathematics And ArtHelp:CategoryCategory:Minimal SurfacesDiscussion About Edits From This IP Address [n]A List Of Edits Made From This IP Address [y]View The Content Page [c]Discussion About The Content Page [t]Edit This Page [e]Visit The Main Page [z]Guides To Browsing WikipediaFeatured Content – The Best Of WikipediaFind Background Information On Current EventsLoad A Random Article [x]Guidance On How To Use And Edit WikipediaFind Out About WikipediaAbout The Project, What You Can Do, Where To Find ThingsA List Of Recent Changes In The Wiki [r]List Of All English Wikipedia Pages Containing Links To This Page [j]Recent Changes In Pages Linked From This Page [k]Upload Files [u]A List Of All Special Pages [q]Wikipedia:AboutWikipedia:General Disclaimer



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