The minimal perimeter for N confined deformable bubbles of equal area

Electronic Journal of Combinatorics

Volumn 17, Issue 1, 2010, Pages 1-25

  Cox, S J ^a   Flikkema, E ^a  

^a ABERYSTWYTH UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77955689706     PISSN: None     EISSN: 10778926     Source Type: Journal    
DOI: 10.37236/317     Document Type: Article

References (42)

1. D. Weaire and S. Hutzler. The Physics of Foams. Clarendon Press, Oxford, 1999.
2. J.A.F. Plateau. Statique Expérimentale Et Théorique Des Liquides Soumis Aux Seules Forces Moléculaires. Gauthier-Villars, Paris, 1873.
3. J.E. Taylor. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. Math., 103:489-539, 1976.
4. W. Thomson. On the Division of Space with Minimum Partitional Area. Phil. Mag., 24:503-514, 1887.
5. D. Weaire, editor. The Kelvin Problem. Taylor & Francis, London, 1994.
6. D. Weaire and R. Phelan. A counter-example to Kelvin's conjecture on minimal surfaces. Phil. Mag. Lett., 69:107-110, 1994.
7. T.C. Hales. The honeycomb conjecture. Discrete Comput. Geom., 25:1-22, 2001.
8. F. Morgan. Geometric Measure Theory: A Beginner's Guide. Academic Press, San Diego, 4th edition, 2009.
9. M.F. Vaz, S.J. Cox, and M.D. Alonso. Minimum energy configurations of small bidisperse bubble clusters. J. Phys.: Condens. Matter, 16:4165-4175, 2004.
10. M. Alfaro, J. Brock, J. Foisy, N. Hodges, and J. Zimba. Compound soap bubbles in the plane, SMALL Geometry Group report, 1990.
11. J. Foisy, M. Alfaro, J. Brock, N. Hodges, and J. Zimba. The standard double soap bubble in R2 uniquely minimizes perimeter. Pac. J. Math., 159:3542-3546, 1993.
12. F. Morgan. Mathematicians, including undergraduates, look at soap bubbles. Amer. Math. Monthly, 101:343-351, 1994.
13. W. Wichiramala. Proof of the planar triple bubble conjecture. J. reine. angew. Math., 567:1-50, 2004.
14. S.J. Cox, F. Graner, M.F. Vaz, C. Monnereau-Pittet, and N. Pittet. Minimal perimeter for N identical bubbles in two dimensions: calculations and simulations. Phil. Mag., 83:1393-1406, 2003.
15. D. Weaire and N. Rivier. Soap, cells and statistics-random patterns in two dimensions. Contemp. Phys., 25:59-99, 1984.
16. S.J. Cox. Calculations of the minimal perimeter for N deformable cells of equal area confined in a circle. Phil. Mag. Letts., 86:569-578, 2006.
17. F. Graner, Y. Jiang, E. Janiaud, and C. Flament. Equilibrium states and ground state of two-dimensional fluid foams. Phys. Rev. E, 63:011402, 2001.
18. S.J. Cox and F. Graner. Large two-dimensional clusters of equal-area bubbles. Phil. Mag., 83:2573-2584, 2003.
19. F. Morgan. Soap Bubble Clusters. Rev. Modern Phys., 79:821-827, 2007.
20. A. Heppes and F. Morgan. Planar Clusters and Perimeter Bounds. Phil. Mag., 85: 1333-1345, 2005.
21. A. Cañete and M. Ritore, Least-perimeter partitions of the disk into three regions of given areas. Indiana Univ. Math. J., 53:883-904, 2004.
22. Y. Tomonaga. Geometry of Length and Area. Dept. of Mathematics, Utsunomiya University, Japan, 1974.
23. M.N. Bleicher. Isoperimetric divisions into several cells with natural boundary. In Intuitive geometry. North-Holland, Amsterdam., 1987. Colloq. Math. Soc. János Bolyai, 48:63-84.
24. E. Baumann, 2007. http://private.mcnet.ch/baumann/EqAreaOverview.htm.
25. 2. Real Analysis Exchange, 22:645-654, 1996/7.
26. M. Engelstein. The Least-Perimeter Partition of a Sphere into Four Equal Areas. Discrete Comput. Geom. (to appear). DOI: 10.1007/s00454-009-9197-8, 2009.
27. T.C. Hales. The honeycomb conjecture on the sphere. arXiv:math/0211234, 2002.
28. F.J. Almgren and J.E. Taylor. Geometry of soap films. Sci. Am., 235:82-93, 1976.
29. B. Shuttleworth. Private communication, 2007.
30. M. Goldberg. The Isoperimetric Problem for Polyhedra. Tohoku Math. J., 40:226-236, 1934.
31. E. Newkirk. Least-Perimeter Partitions of the Sphere. Undergraduate thesis, Williams College, Williamstown, Massachusetts, 2009.
32. T. Hayashi and R.W. Carthew. Surface mechanics mediate pattern formation in the developing retina. Nature, 431:647-652, 2004.
33. T. Tarnai. Optimum packing of circles in a circle. In I. Hargittai and T. Laurent, editors, Symmetry 2000, pages 121-132. Portland, 2002.
34. T. Tarnai and K. Miyazaki. Circle Packings and the Sacred Lotus. Leonardo, 36: 145-150, 2003.
35. C.B. Barber, D.P. Dobkin, and H.T. Huhdanpaa. The Quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software, 22:469-483, 1996.
36. K. Brakke. The Surface Evolver. Exp. Math., 1:141-165, 1992. www.susqu.edu/brakke/.
37. Y.-J. Lai and L. I. Packings and defects of strongly coupled two-dimensional Coulomb clusters:Numerical simulation. Phys. Rev E, 60:4743-4753, 1999.
38. G. Brinkmann, O. Delgado Friedrichs, A. Dress, and T. Harmuth. CaGe - a virtual environment for studying some special classes of large molecules. MATCH Commun. Math. Comput. Chem., 36:233-237, 1997. http://www.mathematik.unibielefeld.de/_CaGe/.
39. D. Ferguson. Private Communication, 2009.
40. B.A. Shuttleworth. Isoperimetric problems in a triangle. Master's thesis, Aberystwyth University, 2008.
41. E.B. Matzke. The three-dimensional shape of bubbles in foam - an analysis of the rôle of surface forces in three-dimensional cell shape determination. Am. J. Botany, 33:58-80, 1946.
42. A.M. Kraynik, D.A. Reinelt, and F. van Swol. The structure of random monodisperse foam. Phys. Rev. E, 67:031403, 2003.