Abstract.
We present conjectured candidates for the least perimeter partition of a disc into \(N \le 10\) connected regions which take one of two possible areas. We assume that the optimal partition is connected and enumerate all three-connected simple cubic graphs for each N. Candidate structures are obtained by assigning different areas to the regions: for even N there are N/2 bubbles of one area and N/2 bubbles of the other, and for odd N we consider both cases, i.e. in which the extra bubble takes either the larger or the smaller area. The perimeter of each candidate structure is found numerically for a few representative area ratios, and then the data is interpolated to give the conjectured least perimeter candidate for all possible area ratios. For each N we determine the ranges of area ratio for which each least perimeter candidate is optimal; at larger N these ranges are smaller, and there are more transitions from one optimal structure to another as the area ratio is varied. When the area ratio is significantly far from one, the least perimeter partitions tend to have a “mixed” configuration, in which bubbles of the same area are not adjacent to each other.
Graphical abstract
![](http://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepje%2Fi2019-11857-0/MediaObjects/10189_2019_11857_Fig1_HTML.jpg)
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Cantat, S. Cohen-Addad, F. Elias, F. Graner, R. Höhler, O. Pitois, F. Rouyer, A. Saint-Jalmes, Foams - Structure and Dynamics (OUP, Oxford, 2013)
D. Weaire, S. Hutzler, The Physics of Foams (Clarendon Press, Oxford, 1999)
D. Weaire, R. Phelan, Philos. Mag. Lett. 69, 107 (1994)
W. Thomson, Philos. Mag. 24, 503 (1887)
F. Morgan, Geometric Measure Theory: A Beginner's Guide, 4th edition (Academic Press, San Diego, 2008)
T.C. Hales, Discret. Comput. Geom. 25, 1 (2001)
W. Wichiramala, J. Reine Angew. Math. 567, 1 (2004)
M. Engelstein, Discret. Comput. Geom. 44, 645 (2010)
M. Hutchings, F. Morgan, M. Ritoré, A. Ros, Ann. Math. 155, 459 (2002)
M.A. Fortes, P.I.C. Teixeira, Eur. Phys. J. E 6, 133 (2001)
M.F. Vaz, S.J. Cox, M.D. Alonso, J. Phys.: Condens. Matter 16, 4165 (2004)
A. Cañete, M. Ritore, Indiana Univ. Math. J. 53, 883 (2004)
Y. Tomonaga, Geometry of Length and Area (Department of Mathematics, Utsunomiya University, Japan, 1974)
M.N. Bleicher, Colloq. Math. Soc. János Bolyai 48, 63 (1987)
S.J. Cox, E. Flikkema, Electron. J. Comb. 17, R45 (2010)
B. Bogosel, E. Oudet, Exp. Math. 26, 496 (2016)
P.I.C. Teixeira, F. Graner, M.A. Fortes, Eur. Phys. J. E 9, 161 (2002)
S.J. Cox, J. Non-Newton. Fluid Mech. 137, 39 (2006)
C. Quilliet, S. Ataei Talebi, D. Rabaud, J. Käfer, S.J. Cox, F. Graner, Philos. Mag. Lett. 88, 651 (2008)
M.F. Vaz, S.J. Cox, P.I.C. Teixeira, Philos. Mag. 91, 4345 (2011)
J.A.F. Plateau, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires (Gauthier-Villars, Paris, 1873)
J.E. Taylor, Ann. Math. 103, 489 (1976)
F. Morgan, Pac. J. Math. 165, 347 (1994)
G. Brinkmann, O. Delgado Friedrichs, A. Dress, T. Harmuth, MATCH Commun. Math. Comput. Chem. 36, 233 (1997) http://www.mathematik.uni-bielefeld.de/~CaGe/
K. Brakke, Exp. Math. 1, 141 (1992)
A.M. Kraynik, D.A. Reinelt, F. van Swol, Phys. Rev. Lett. 93, 208301 (2004)
S.J. Cox, Philos. Mag. Lett. 86, 569 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Headley, F., Cox, S. Least perimeter partition of the disc into N bubbles of two different areas. Eur. Phys. J. E 42, 92 (2019). https://doi.org/10.1140/epje/i2019-11857-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2019-11857-0