Abstract
We consider mathematical models of bubbles, foams and froths, as collections of surfaces which minimize area under volume constraints. The resulting surfaces have constant mean curvature and an invariant notion of equilibrium forces. The possible singularities are described by Plateau’s rules; this means that combinatorially a foam is dual to some triangulation of space. We examine certain restrictions on the combinatorics of triangulations and some useful ways to construct triangulations. Finally, we examine particular structures, like the family of tetrahedrally close-packed structures. These include the one used by Weaire and Phelan in their counterexample to the Kelvin conjecture, and they all seem useful for generating good equal-volume foams.
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References
Alexandrov, A. D.: 1958, `Uniqueness Theorems for Surfaces in the Large, I’. Vestnik Leningrad Univ. Math. 19(13), 5–8. English transi. in Amer. Math. Soc. Transi. (Ser. 2) 21 (1962), 412–416.
Almgren, Jr., F. J.: 1976, `Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints’. Mem. Amer. Math. Soc. 4 (165).
Aste, T., D. Boosé, and N. Rivier: 1996, `From One Cell to the Whole Froth: A Dynamical Map’. Phys. Rev. 53, 6181–6191.
Brakke, K. and F. Morgan: 1996, `Instability of the Wet X Soap Film’. Preprint.
Brakke, K. A.: 1992, `The Surface Evolver’. Exper. Math. 1 (2), 141–165.
Brakke, K. A. and J. M. Sullivan: 1997, `Using Symmetry Features of the Surface Evolver to Study Foams’. In: K. Polthier and H.-C. Hege (eds.): Visualization andMathematics. Heidelberg, pp. 95–117.
Choe, J.: 1989, `On the Existence and Regularity of Fundamental Domains with Least Boundary Area’. J. Diff. Geom. 29, 623–663.
Coxeter, H. S. M.: 1958, `Close-Packing and Froth’. Ill. J. Math. 2(4B), 746–758. Reprinted in [52].
Delaunay, C.: 1841, `Sur la surface de révolution, dont la courbure moyenne est constante’. Journal de mathématiques 6, 309–320.
Dierkes, U., S. Hildebrandt, A. Küster, and O. Wohlrab: 1992, Minimal Surfaces I, Vol. 295 of Grundlehren der Mathematischen Wissenschaften. Berlin: Springer-Verlag.
Frank, F. C. and J. S. Kasper: 1958, `Complex Alloy Structures Regarded as Sphere Packings. I. Definitions and Basic Principles’. Acta Crystall. 11, 184–190.
Frank, F. C. and J. S. Kasper: 1959, `Complex Alloy Structures Regarded as Sphere Packings. II. Analysis and Classification of Representative Structures’. Acta Crystall. 12, 483–499.
Große-Brauckmann, K., R. Kusner, and J. M. Sullivan: 1997, `Classification of Embedded Constant Mean Curvature Surfaces with Genus Zero and Three Ends’. GANG Preprint IV.29, UMass.
Große-Brauckmann, K., R. Kusner, and J. M. Sullivan: 1998, `Constant Mean Curvature Surfaces with Cylindrical Ends’. To appear in the Springer proceedings of VisMath’97.
Hass, J., M. Hutchings, and R. Schlafiy: 1995, `The Double Bubble Conjecture’. Electron. Res. Announc. Amer. Math. Soc. 1 (3), 98–102.
Hildebrandt, S.: 1970, `On the Plateau problem for surfaces of constant mean curvature’. Comm. Pure Appl. Math. 23, 97–114.
Hildebrandt, S. and A. Tromba: 1996, The Parsimonious Universe. New York: Copernicus.
Hoffman, D. and W. H. Meeks, III: 1990, `Embedded Minimal Surfaces of Finite Topology’. Ann. of Math. 131 (1), 1–34.
Hsu, L., R. Kusner, and J. M. Sullivan: 1992, `Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms’. Experimental Mathematics 1(3), 191–207.
Jülicher, F., U. Seifert, and R. Lipowsky: 1993, `Conformal Degeneracy and Conformal Diffusion of Vesicles’. Phys. Rev. Lett. 71, 452–455.
Korevaar, N., R. Kusner, and B. Solomon: 1989, `The Structure of Complete Embedded Surfaces with Constant Mean Curvature’. J. Di, Q Geom. 30, 465–503.
Kraynik, A. M., R. Kusner, R. Phelan, and J. M. Sullivan, ‘TCP Structures as Equal-Volume Foams’ In preparation.
Kraynik, A. M. and D. A. Reinelt: 1996, `Elastic-Plastic Behavior of a Kelvin Foam’. Forma 11(3), 255–270. Reprinted in [52].
Kusner, R.: 1992, `The Number of Faces in a Minimal Foam’. Proc. R. Soc. Lond. 439, 683–686.
Kusner, R. and N. Schmitt: 1996, `On the Spinor Representation of Minimal Surfaces’. GANG preprint 11I.27, UMass.
Kusner, R. and J. M. Sullivan: 1996, `Comparing the Weaire-Phelan Equal-Volume Foam to Kelvin’s Foam’. Forma 11(3), 233–242. Reprinted in [52].
Luo, F. and R. Stong: 1993, ‘Combinatorics of Triangulations of 3-Manifolds’. Trans. Amer. Math. Soc. 337 (2), 891–906.
Matzke, E. B.: 1946, `The Three-Dimensional Shape of Bubbles in Foam’. Amer. J. Botany 33, 58–80.
Meier, W. M. and D. H. Olson: 1992, Atlas of Zeolite Structure Types. Butterworths, 3rd edition.
Michalet, X. and D. Bensimon: 1995, `Observations of Stable Shapes and Conformal Diffusion in Genus 2 Vesicles’. Science 269, 666–668.
Morgan, F.: 1994, `Clusters Minimizing Area Plus Length of Singular Curves’. Math. Ann. 299 (4), 697–714.
Morgan, F.: 1995a, A Beginner’s Guide to Geometric Measure Theory. Academic Press, 2nd edition.
Morgan, F.: 1995b, F.: 1995b, `The Double Soap Bubble Conjecture’. MAA FOCUS pp. 6–7. Dec. 1995.
Morgan, F.: 1996, `The Hexagonal Honeycomb Conjecture’. Preprint.
Okabe, A., B. Boots, and K. Sugihara: 1992, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley & Sons.
O’Keeffe, M.: 1997, `Crystal Structures as Periodic Foams and vice versa’. Appearing in this volume.
O’Keeffe, M. and B. G. Hyde: 1996, Crystal Structures I: Patterns and Symmetry. Washington: Mineral Soc. Amer.
Osserman, R.: 1986, A Survey of Minimal Surfaces. New York: Dover Publications, 2nd edition.
Plateau, J.: 1873, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires. Paris: Gauthier-villars.
Reinelt, D. A. and A. M. Kraynik: 1993, `Large Elastic Deformations of Three-Dimensional Foams and Highly Concentrated Emulsions’. J. of Colloid and Interface Science 159, 460–470.
Rivier, N.: 1994, `Kelvin’s Conjecture on Minimal Froths and the Counter-Example of Weaire and Phelan’. Europhys. Lett. 7 (6), 523–528.
Rogers, C. A.: 1958, `The Packing of Equal Spheres’. Proc. London Math. Soc. 8, 609–620.
Sadoc, J.-F. and R. Mosseri: 1982, `Order and Disorder in Amorphous Tetrahedrally Coordintaed Semiconductors: A Curved-Space Description’. Philos. Mag. 45, 467.
Schwarz, H. A.: 1884, ‘Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt, als jeder andere Körper gleichen Volumnes’. Nach. Ges. Wiss. Göttingen pp. 1–13. Reprinted in 1972 in Gesammelte mathematische Abhandlungen, pp. II. 327–340, New York: Chelsea.
Senechal, M.: 1990, Crystalline Symmetries. Adam Hilger.
Shoemaker, D. P. and C. B. Shoemaker: 1986, `Concerning the Relative Numbers of Atomic Coordination Types in Tetrahedrally Close Packed Metal Structures’. Acta Crystall. 42, 3–11.
Sullivan, J. M.: 1988, `The vcs Software for Computing Voronoi Diagrams’. Available by email from jmstmath.uiuc.edu.
Sullivan, J. M. and F. Morgan (Editors): 1996, `Open Problems in Soap Bubble Geometry: Posed at the Burlington Mathfest in August 1995’. International J. of Math. 7 (6), 833–842.
Taylor, J. E.: 1976, `The Structure of Singularities in Soap-Bubble-Like and SoapFilm-Like Minimal Surfaces’. Ann. of Math. 103, 489–539.
Thompson, Sir W. (Lord Kelvin): 1887, `On the Division of Space with Minimum Partitional Area’. Philos. Mag. 24, 503–514. Also published in Acta Math. 11, 121–134, and reprinted in [52].
Thurston, W. P.: 1997, Three-Dimensional Geometry and Topology, Vol. 1. Princeton. Edited by Silvio Levy.
Weaire, D. (ed.): 1997, The Kelvin Problem. Taylor & Francis.
Weaire, D. and R. Phelan: 1994, `A Counter-Example to Kelvin’s Conjecture on Minimal Surfaces’. Phil. Mag. Lett. 69(2), 107–110. Reprinted in [52].
Williams, R. E.: 1968, `Space Filling Polyhedron: Its Relation to Aggregates of Soap Bubbles, Plant Cells, and Metal Crystallites’. Science 161, 276–277.
Willmore, T. J.: 1992, `A Survey on Willmore Immersions’. In: Geometry and Topology of Submanifolds, IV (Leuven, 1991 ). pp. 11–16.
Wintz, W., H.-G. Döbereiner, and U. Seifert: 1996, `Starfish Vesicles’. Europhys. Lett. 33, 403–408.
Yarmolyuk, Y. P. and P. I. Kripyakevich: 1974. Kristallographiya 19, 539–545. Translated in Soy. Phys. Crystallogr. 19, 334–337.
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Sullivan, J.M. (1999). The Geometry of Bubbles and Foams. In: Sadoc, J.F., Rivier, N. (eds) Foams and Emulsions. NATO ASI Series, vol 354. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9157-7_23
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DOI: https://doi.org/10.1007/978-94-015-9157-7_23
Publisher Name: Springer, Dordrecht
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