DOCSLIB.ORG

Energy Minimizers and the Beauty of Soap Bubbles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Energy Minimizers and the Beauty of Soap Bubbles Energy Minimizers and the Beauty of Soap Bubbles Sonia Kovalevsky High School Day October 28, 2011 Soap Films & Soap Bubbles: mathematics, biology, chemistry, physics… architecture • Scientists have been experimenting with soap bubbles and films for hundreds of years • A = area of a soap film contained by a fixed boundary σ f = film tension (constant in many cases) then energy available to alter the film shape is given by F = σ f A free energy € • More in general, total energy necessary to increase the area of a soap film from zero to A: A F = ∫ σ f dA € 0 • Soap film is in equilibrium when free energy has a local minimum • Conclusion: soap films in equilibrium will have minimum area € …soap films in equilibrium will have minimum area… Try it! Plateau Problem: Find a minimal surface with a given boundary • Empirical solutions to this problem may be obtained by dipping a model of the boundary into soapy water Minimum surface contained by the edges of a tetrahedron """" # $ Minimum surface formed by the twelve edges of a cubic framework • Also, when we blow a soap bubble, the soapy surface stretches, when we stop the film tends toward equilibrium, and will form a minimal surface Bubbles at work! • Isoperimetric property: Circumferences are the least-perimeter way to enclose a given surface area • Least-area way to enclose a given volume of air: sphere We see this all the time! When we blow into a soapy film, we fix a volume of air (the air that we blow), while the bubble soap forms the spherical surface which solves the minimization problem! • Least-area way to enclose and separate two given volumes of air? ? Double Bubble Conjecture: Is the familiar double soap bubble the solution? TRUE! Different Volumes Same Volumes (Proved by an international team of mathematicians on 2000) J. Plateau discovered experimentally over hundred years ago that soap films contained by a framework always satisfy three geometrical conditions: • Three smooth surfaces of a soap film intersect along a line • The angle between any two tangent planes to the intersecting surfaces, at any point along the line of intersection of three surface is 1200 • Four of the lines, each formed by the intersection of three surfaces, meet at a point and the angle between any pair of adjacent lines is 1090 28’ (angle whose cosine is -1/3) Only in 1976, Jean E. Taylor was able to prove mathematically that the laws of Plateau are correct! Plateau’s principles at work! Plateau’s principles at work! Plateau’s principles at work! • The Motorway Problem: what is the shortest path between four (or more) points? For example, how could we link a number of towns by the shortest road, water pipe, or cable? Motorway Problem: possible solutions… (a) (d) & (e) same total length (d) (e) (b) … shorter and shorter… but none is the smallest! (c) (f) Look at soap films! Notice the 1200 angles! Experiments with soap bubbles have inspired also architects, notably German architect Otto Frei • Montreal, Canada: German Pavilion, Expo ’67 Architect: Frei Otto Soap bubbles models were carefully constructed, photographed and measured. Solid models were then build and tested in wind tunnels • Munich, Germany: Olympic Games, 1972 Designed by Gunter Behnisch, with Otto Frei as roof design consultant • Beijing National Aquatics Center, 2008 References • Cyril Isenberg, The Science of Soap Films and Soap Bubbles, Dover Publications, 1992 • Michele Emmer, Architecture and Mathematics: Soap Bubbles and Soap Films, pp. 53-65 in Nexus: Architecture and Mathematics, ed. Kim Williams, Edizioni dell’Erba, 1996 • Sami Polatoz, From Soap Bubbles to Technology, The Fountain Magazine, November-December 2008, Issue 66 • Frank Morgan, Double Bubble No More Trouble, Math Horizons, November 2000 • John Sullivan, http://torus.math.uiuc.edu/jms/Images/ • David Lovett, Demonstrating Science with Soap Films, Institute of Physics Publishing, 1994 • http://www.flickr.com/photos/scottgr/376657900/sizes/m/in/photostream/ • http://www.arch.mcgill.ca/prof/sijpkes/arch374/winter2001/tal/soapfilm/ final.htm • http://www.best-of-munich.com/olympic-park/img/munich-olympic-park.jpg • http://www.westland.net/expo67/map-docs/images/germany3.jpg • Beijing National Aquatic Center picture from Wikipedia .
Load more

Recommended publications
  • Governing Equations for Simulations of Soap Bubbles
    University of California Merced Capstone Project Governing Equations For Simulations Of Soap Bubbles Author: Research Advisor: Ivan Navarro Fran¸cois Blanchette 1 Introduction The capstone project is an extension of work previously done by Fran¸coisBlanchette and Terry P. Bigioni in which they studied the coalescence of drops with either a hori- zontal reservoir or a drop of a different size (Blanchette and Bigioni, 2006). Specifically, they looked at the partial coalescence of a drop, which leaves behind a smaller "daugh- ter" droplet due to the incomplete merging process. Numerical simulations were used to study coalescence of a drop slowly coming into contact with a horizontal resevoir in which the fluid in the drop is the same fluid as that below the interface (Blanchette and Bigioni, 2006). The research conducted by Blanchette and Bigioni starts with a drop at rest on a flat interface. A drop of water will then merge with an underlying resevoir (water in this case), forming a single interface. Our research involves using that same numerical ap- proach only this time a soap bubble will be our fluid of interest. Soap bubbles differ from water drops on the fact that rather than having just a single interface, we now have two interfaces to take into account; the air inside the soap bubble along with the soap film on the boundary, and the soap film with any other fluid on the outside. Due to this double interface, some modifications will now be imposed on the boundary conditions involving surface tension along the interface. Also unlike drops, soap bubble thickness is finite which will mean we must keep track of it.
    [Show full text]
  • Weighted Transparency: Literal and Phenomenal
    Frei Otto soap bubble model (top left), Antonio Gaudi gravity model (bottom left), Miguel Fisac fabric form concrete wall (right) Weighted Transparency: Literal and Phenomenal SPRING 2020, OPTION STUDIO VISITING ASSOCIATE PROFESSOR NAOMI FRANGOS “As soon as we adventure on the paths of the physicist, we learn to weigh and to measure, to deal with time and space and mass and their related concepts, and to find more and more our knowledge expressed and our needs satisfied through the concept of number, as in the dreams and visions of Plato and Pythagoras; for modern chemistry would have gladdened the hearts of those great philosophic dreamers.” On Growth and Form, D’Arcy Wentworth Thompson Today’s maker architect is such a dreamt physicist. What is at stake is the ability to balance critical form-finding informed by the behavior of matter itself with the rigor of precision afforded by computational thinking. This studio transposes ideas of cross- disciplinarity in design, art, science, engineering and material studies pioneered by our forerunner master-builders into built prototypes by studying varying degrees of literal and phenomenal transparency achieved through notions of weight. Working primarily with a combination of plaster/concrete and glass/porcelain, juxtapositions are provoked between solid and void, heavy and light, opaque and transparent, smooth and textured, volume and surface. How can the actual weight of a material affect its sense of mass? How can mass capture the phenomena of weightlessness? How can physical material properties dictate appearances beyond optical transparency? Using dynamic matter/flexible formwork (i.e. salt, sand, gravel, fabric), suspended/submerged and compression/ expansion systems, weight plays a major role in deriving methods of fabrication and determining experiential qualities of made artifacts.
    [Show full text]
  • Universität Regensburg Mathematik
    UniversitÄatRegensburg Mathematik Parametric Approximation of Surface Clusters driven by Isotropic and Anisotropic Surface Energies J.W. Barrett, H. Garcke, R. NÄurnberg Preprint Nr. 04/2009 Parametric Approximation of Surface Clusters driven by Isotropic and Anisotropic Surface Energies John W. Barrett† Harald Garcke‡ Robert N¨urnberg† Abstract We present a variational formulation for the evolution of surface clusters in R3 by mean curvature flow, surface diffusion and their anisotropic variants. We intro- duce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully dis- crete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, in- cluding isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including for regularized crystalline surface energy densities. Key words. surface clusters, mean curvature flow, surface diffusion, soap bubbles, triple junction lines, parametric finite elements, anisotropy, tangential movement AMS subject classifications. 65M60, 65M12, 35K55, 53C44, 74E10, 74E15 1 Introduction Equilibrium soap bubble clusters are stationary solutions of the variational problem in which one seeks a least area way to enclose and separate a number of regions with pre- scribed volumes. The relevant energy in this case is given as the sum of the total surface area.
    [Show full text]
  • Vibrations of Fractal Drums
    VOLUME 67, NUMBER 21 PHYSICAL REVIEW LETTERS 18 NOVEMBER 1991 Vibrations of Fractal Drums B. Sapoval, Th. Gobron, and A. Margolina "' Laboratoire de Physique de la Matiere Condensee, Ecole Polytechnique, 91128 Palaiseau CEDEX, France (Received l7 July l991) Fractal boundary conditions drastically alter wave excitations. The low-frequency vibrations of a membrane bounded by a rigid fractal contour are observed and localized modes are found. The first lower eigenmodes are computed using an analogy between the wave and the diffusion equations. The fractal frontier induces a strong confinement of the wave analogous to superlocalization. The wave forms exhibit singular derivatives near the boundary. PACS numbers: 64.60.Ak, 03.40.Kf, 63.50.+x, 71.55.3v Objects with irregular geometry are ubiquitous in na- shown in Fig. 2. ture and their vibrational properties are of general in- The observation of confined modes was a surprise be- terest. For instance, the dependence of sea waves on the cause, in principle, the symmetry of the structure forbids topography of coastlines is a largely unanswered question. their existence. In fact, this experimental localization is The emergence of fractal geometry was a significant due both to damping and to the existence of narrow paths breakthrough in the description of irregularity [11. It has in the geometry of the membrane. If only one of the been suggested that the very existence of some of the equivalent regions like A is excited, then a certain time T, fractal structures found in nature is attributable to a called the delocalization time is necessary for the excita- self-stabilization due to their ability to damp harmonic tion to travel from A to B.
    [Show full text]
  • Memoirs of a Bubble Blower
    Memoirs of a Bubble Blower BY BERNARD ZUBROVVSKI Reprinted from Technology Review, Volume 85, Number 8, Nov/Dec 1982 Copyright 1982, Alumni Association of the Massachusetts Institute of Technolgy, Cambridge, Massachusetts 02139 EDUCATION A SPECIAL REPORT Memoirs N teaching science to children, I have dren go a step further by creating three- found that the best topics are those dimensional clusters of bubbles that rise I that are equally fascinating to chil• up a plexiglass tube. These bubbles are dren and adults alike. And learning is not as regular as those in the honeycomb most enjoyable when teacher and students formation, but closer observation reveals are exploring together. Blowing bubbles that they have a definite pattern: four provides just such qualities. bubbles (and never more than four) are Bubble blowing is an exciting and fruit• usually in direct contact with one another, ful way for children to develop basic in• with their surfaces meeting at a vertex at tuition in science and mathematics. And angles of about 109 degrees. And each working with bubbles has prompted me to of a bubble in the cluster will have roughly the wander into various scientific realms such same number of sides, a fact that turns out as surface physics, cellular biology, topol• to be scientifically significant. ogy, and architecture. Bubbles and soap Bubble In the 1940s, E.B. Matzke, a well- films are also aesthetically appealing, and known botanist who experimented with the children and I have discovered bubbles bubbles, discovered that bubbles in an to be ideal for making sculptures. In fact, array have an average of 14 sides.
    [Show full text]
  • Computer Graphics and Soap Bubbles
    Computer Graphics and Soap Bubbles Cheung Leung Fu & Ng Tze Beng Department of Mathematics National University of Singapore Singapore 0511 Soap Bubbles From years of study and of contemplation A boy, with bowl and straw, sits and blows, An old man brews a work of clarity, Filling with breath the bubbles from the bowl. A gay and involuted dissertation Each praises like a hymn, and each one glows; Discoursing on sweet wisdom playfully Into the filmy beads he blows his soul. An eager student bent on storming heights Old man, student, boy, all these three Has delved in archives and in libraries, Out of the Maya.. foam of the universe But adds the touch of genius when he writes Create illusions. None is better or worse A first book full of deepest subtleties. But in each of them the Light of Eternity Sees its reflection, and burns more joyfully. The Glass Bead Game Hermann Hesse The soap bubble, a wonderful toy for children, a motif for poetry, is at the same time a mathematical entity that has a fairly long history. Thanks to the advancement of computer graphics which led to solutions of decade-old conjectures, 'soap bubbles' have become once again a focus of much mathematical research in recent years. Anyone who has seen or played with a soap bubble knows that it is a spherical object. It is indeed a sphere if one defines a soap bubble to be a surface minimizing the area among all surfaces enclosing the same given volume. Relaxing this condition, mathematicians defined a soap bubble or surface of constant mean curvature to be a surface which is locally the stationary point of the area function when compared again locally to neighbouring surfaces enclosing the same volume.
    [Show full text]
  • Graphs in Nature
    Graphs in Nature David Eppstein University of California, Irvine Algorithms and Data Structures Symposium (WADS) August 2019 Inspiration: Steinitz's theorem Purely combinatorial characterization of geometric objects: Graphs of convex polyhedra are exactly the 3-vertex-connected planar graphs Image: Kluka [2006] Overview Cracked surfaces, bubble foams, and crumpled paper also form natural graph-like structures What properties do these graphs have? How can we recognize and synthesize them? I. Cracks and Needles Motorcycle graphs: Canonical quad mesh partitioning Problem: partition irregular quad-mesh into regular submeshes [Eppstein et al. 2008] Inspiration: Light cycle game from TRON movies Mesh partitioning method Grow cut paths outwards from each irregular (non-degree-4) vertex Cut paths continue straight across regular (degree-4) vertices They stop when they run into another path Result: approximation to optimal partition (exact optimum is NP-complete) Mesh-free motorcycle graphs Earlier... Motorcycles move from initial points with given velocities When they hit trails of other motorcycles, they crash [Eppstein and Erickson 1999] Application of mesh-free motorcycle graphs Initially: A simplified model of the inward movement of reflex vertices in straight skeletons, a rectilinear variant of medial axes with applications including building roof construction, folding and cutting problems, surface interpolation, geographic analysis, and mesh construction Later: Subroutine for constructing straight skeletons of simple polygons [Cheng and Vigneron 2007; Huber and Held 2012] Image: Huber [2012] Construction of mesh-free motorcycle graphs Main ideas: Define asymmetric distance: Time when one motorcycle would crash into another's trail Repeatedly find closest pair and eliminate crashed motorcycle Image: Dancede [2011] O(n17=11+) [Eppstein and Erickson 1999] Improved to O(n4=3+) [Vigneron and Yan 2014] Additional log speedup using mutual nearest neighbors instead of closest pairs [Mamano et al.
    [Show full text]
  • On the Shape of Giant Soap Bubbles
    On the shape of giant soap bubbles Caroline Cohena,1, Baptiste Darbois Texiera,1, Etienne Reyssatb,2, Jacco H. Snoeijerc,d,e, David Quer´ e´ b, and Christophe Claneta aLaboratoire d’Hydrodynamique de l’X, UMR 7646 CNRS, Ecole´ Polytechnique, 91128 Palaiseau Cedex, France; bLaboratoire de Physique et Mecanique´ des Milieux Het´ erog´ enes` (PMMH), UMR 7636 du CNRS, ESPCI Paris/Paris Sciences et Lettres (PSL) Research University/Sorbonne Universites/Universit´ e´ Paris Diderot, 75005 Paris, France; cPhysics of Fluids Group, University of Twente, 7500 AE Enschede, The Netherlands; dMESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands; and eDepartment of Applied Physics, Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved January 19, 2017 (received for review October 14, 2016) We study the effect of gravity on giant soap bubbles and show The experimental setup dedicated to the study of such large 2 that it becomes dominant above the critical size ` = a =e0, where bubbles is presented in Experimental Setup, before information p e0 is the mean thickness of the soap film and a = γb/ρg is the on Experimental Results and Model. The discussion on the asymp- capillary length (γb stands for vapor–liquid surface tension, and ρ totic shape and the analogy with inflated structures is presented stands for the liquid density). We first show experimentally that in Analogy with Inflatable Structures. large soap bubbles do not retain a spherical shape but flatten when increasing their size. A theoretical model is then developed Experimental Setup to account for this effect, predicting the shape based on mechan- The soap solution is prepared by mixing two volumes of Dreft© ical equilibrium.
    [Show full text]
  • Soap Films: Statics and Dynamics
    Soap Films: Statics and Dynamics Frederik Brasz January 8, 2010 1 Introduction Soap films and bubbles have long fascinated humans with their beauty. The study of soap films is believed to have started around the time of Leonardo da Vinci. Since then, research has proceeded in two distinct directions. On the one hand were the mathematicians, who were concerned with finding the shapes of these surfaces by minimizing their area, given some boundary. On the other hand, physical scientists were trying to understand the properties of soap films and bubbles, from the macroscopic behavior down to the molecular description. A crude way to categorize these two approaches is to call the mathematical shape- finding approach \statics" and the physical approach \dynamics", although there are certainly problems involving static soap films that require a physical description. This is my implication in this paper when I divide the discussion into statics and dynamics. In the statics section, the thickness of the film, concentration of surfactants, and gravity will all be quickly neglected, as the equilibrium shape of the surface becomes the only matter of importance. I will explain why for a soap film enclosing no volume, the mean curvature must everywhere be equal to zero, and why this is equivalent to the area of the film being minimized. For future reference, a minimal surface is defined as a two-dimensional surface with mean curvature equal to zero, so in general soap films take the shape of minimal surfaces. Carrying out the area minimization of a film using the Euler-Lagrange equation will result in a nonlinear partial differential equation known as Lagrange's Equation, which all minimal surfaces must satisfy.
    [Show full text]
  • Chemomechanical Simulation of Soap Film Flow on Spherical Bubbles
    Chemomechanical Simulation of Soap Film Flow on Spherical Bubbles WEIZHEN HUANG, JULIAN ISERINGHAUSEN, and TOM KNEIPHOF, University of Bonn, Germany ZIYIN QU and CHENFANFU JIANG, University of Pennsylvania, USA MATTHIAS B. HULLIN, University of Bonn, Germany Fig. 1. Spatially varying iridescence of a soap bubble evolving over time (left to right). The complex interplay of soap and water induces a complex flowonthe film surface, resulting in an ever changing distribution of film thickness and hence a highly dynamic iridescent texture. This image was simulated using the method described in this paper, and path-traced in Mitsuba [Jakob 2010] using a custom shader under environment lighting. Soap bubbles are widely appreciated for their fragile nature and their colorful In the computer graphics community, it is now widely understood appearance. The natural sciences and, in extension, computer graphics, have how films, bubbles and foams form, evolve and break. On the ren- comprehensively studied the mechanical behavior of films and foams, as dering side, it has become possible to recreate their characteristic well as the optical properties of thin liquid layers. In this paper, we focus on iridescent appearance in physically based renderers. The main pa- the dynamics of material flow within the soap film, which results in fasci- rameter governing this appearance, the thickness of the film, has so nating, extremely detailed patterns. This flow is characterized by a complex far only been driven using ad-hoc noise textures [Glassner 2000], coupling between surfactant concentration and Marangoni surface tension. We propose a novel chemomechanical simulation framework rooted in lu- or was assumed to be constant.
    [Show full text]
  • Cannonballs and Honeycombs, Volume 47, Number 4
    fea-hales.qxp 2/11/00 11:35 AM Page 440 Cannonballs and Honeycombs Thomas C. Hales hen Hilbert intro- market. “We need you down here right duced his famous list of away. We can stack the oranges, but 23 problems, he said we’re having trouble with the arti- a test of the perfec- chokes.” Wtion of a mathe- To me as a discrete geometer Figure 1. An matical problem is whether it there is a serious question be- optimal can be explained to the first hind the flippancy. Why is arrangement person in the street. Even the gulf so large between of equal balls after a full century, intuition and proof? is the face- Hilbert’s problems have Geometry taunts and de- centered never been thoroughly fies us. For example, what cubic tested. Who has ever chatted with about stacking tin cans? Can packing. a telemarketer about the Riemann hy- anyone doubt that parallel rows pothesis or discussed general reciprocity of upright cans give the best arrange- laws with the family physician? ment? Could some disordered heap of cans Last year a journalist from Plymouth, New waste less space? We say certainly not, but the Zealand, decided to put Hilbert’s 18th problem to proof escapes us. What is the shape of the cluster the test and took it to the street. Part of that prob- of three, four, or five soap bubbles of equal vol- lem can be phrased: Is there a better stacking of ume that minimizes total surface area? We blow oranges than the pyramids found at the fruit stand? bubbles and soon discover the answer but cannot In pyramids the oranges fill just over 74% of space prove it.
    [Show full text]
  • Introduction to Nonlinear Geometric Pdes
    Introduction to nonlinear geometric PDEs Thomas Marquardt January 16, 2014 ETH Zurich Department of Mathematics Contents I. Introduction and review of useful material 1 1. Introduction 3 1.1. Scope of the lecture . .3 1.2. Accompanying books . .4 1.3. A historic survey . .5 2. Review: Differential geometry 7 2.1. Hypersurfaces in Rn ..............................7 2.2. Isometric immersions . 11 2.3. First variation of area . 12 3. Review: Linear PDEs of second order 15 3.1. Elliptic PDEs in H¨older spaces . 15 3.2. Elliptic PDEs in Sobolev spaces . 18 3.3. Parabolic PDEs in H¨older spaces . 20 II. Nonlinear elliptic PDEs of second order 25 4. General theory for quasilinear problems 27 4.1. Fixed point theorems: From Brouwer to Leray-Schauder . 27 4.2. Reduction to a priori estimates in the C1,β-norm . 29 4.3. Reduction to a priori estimates in the C1-norm . 30 5. The prescribed mean curvature problem 33 5.1. C0-estimate . 33 5.2. Interior gradient estimate . 37 5.3. Boundary gradient estimate . 40 5.4. Existence and uniqueness theorem . 45 6. General theory for fully nonlinear problems 49 6.1. Fully nonlinear Dirichlet problems . 50 6.2. Fully nonlinear oblique derivative problems . 52 7. The capillary surface problem 57 7.1. C0-estimate . 58 7.2. Global gradient estimate . 60 7.3. Existence and uniqueness theorem . 66 III. Geometric evolution equations 67 8. Classical solutions of MCF and IMCF 69 8.1. Short-time existence . 69 8.2. Evolving graphs under mean curvature flow . 74 8.3.
    [Show full text]