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Least Perimeter Partition of the Disc Into N Bubbles of Two Different Areas Eur. Phys. J. E (2019) 42:92 THE EUROPEAN DOI 10.1140/epje/i2019-11857-0 PHYSICAL JOURNAL E Regular Article Least perimeter partition of the disc into N bubbles of two different areas Francis Headley and Simon Coxa Department of Mathematics, Aberystwyth University, Aberystwyth, SY23 3BZ, UK Received 4 April 2019 and Received in final form 11 June 2019 Published online: 19 July 2019 c The Author(s) 2019. This article is published with open access at Springerlink.com Abstract. We present conjectured candidates for the least perimeter partition of a disc into N ≤ 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. 1 Introduction tions of bidisperse bubble clusters (which, to a good ap- proximation, minimize their surface area [1]) were found Due to their structural stability and low material cost, with the least perimeter configuration [11]. energy-minimizing structures have a wide array of appli- Minimal perimeter partitions of domains with a fixed cations [1, 2]. An example is the Beijing Aquatics Centre, boundary have also generated interest, for example a proof which uses slices of the Weaire-Phelan foam structure [3] of the optimal partition of the disc into N =3re- to create a lightweight and strong but beautiful piece of gions of given areas [12], and many numerical conjectures, architecture. e.g. [13–16]. Such results may lead to further aesthetically The Weaire-Phelan foam is a solution to the cele- pleasing structures like the Water Cube but that are truly brated Kelvin problem, which seeks the minimum sur- foam-like, including their boundary, rather than being un- face area partition of space into bubbles of equal vol- physical sections through a physical object. ume [4]. This builds upon the well-known isoperimetric It would be useful to seek patterns within the class problem concerning the least perimeter shape enclosing a of conjectured minimizers, so that we might be able to given area [5]. Extending this idea to many bubbles with make general statements about the arrangement of re- equal areas has led to further rigorous results for optimal gions within the domain. For example, do smaller bubbles structures, for example the proof of the honeycomb con- tend to cluster together, since a small bubble adjacent to jecture [6], the optimality of the standard triple bubble in a large bubble might deform the larger bubble, increasing the plane [7] and of the tetrahedral partition of the surface its perimeter? How does any such clustering depend on the of the sphere into four regions [8]. differences in areas between bubbles? Fortes et al. [10,17] If the areas of the bubbles are allowed to be unequal, considered partitions of the plane into bubbles of two dif- then the problem of seeking the configuration of least ferent areas and estimated the perimeter “cost” of placing perimeter is more difficult. For N = 2 regions in R3,the bubbles of one area next to another. They predict that double bubble conjecture has been proved [9], and, in the mixed small and large bubbles are favourable when the plane, the extension of the honeycomb to two different difference in areas is large [17] while when the two ar- areas (bidisperse) has led to conjectured solutions [10]. eas are similar the bubbles of each size tend to segregate, There has also been some experimental work that sought a so-called “sorted” configuration. Mechanical agitation to correlate the frequency with which different configura- often “shuffles” bidisperse foams towards such perimeter- minimizing configurations [18–20]. a e-mail: [email protected] Page 2 of 7 Eur. Phys. J. E (2019) 42:92 Fig. 2. A simple cubic three-connected planar graph with three bubbles of equal area, and its associated minimal perimeter (a) P =6.304 (b) P =6.272 monodisperse partition of the disc. Fig. 1. The two different partitions of the disc into N =5 bubbles of equal area. The structure on the right has least The assumption of planarity is natural, since these perimeter P . graphs must be embeddable in the 2D disc. The as- sumption that the graphs are three-regular follows from Plateau’s laws. We assume that the graphs are simple and In this work we seek to generate and test, in a sys- three-connected because any two edges sharing two ver- tematic way, candidate partitions of domains with fixed tices can be decomposed into a configuration with lower boundary, and to examine the optimal patterns found to perimeter P . An example is shown in fig. 3: moving the support conjectures about mixed and sorted configura- lens-shaped bubble to the edge of the disc results in a tions. change in topology and a reduction in perimeter. A sim- Due to the complexity, and in particular the large ilar reduction in perimeter can be achieved in structures number of candidates, we restrict ourselves to a two- with more bubbles by moving a lens towards a threefold dimensional (2D) problem. Thus, we enumerate all par- vertex and performing the same change in topology. titions of a disc and evaluate the perimeter of each one to Our conjectured minimizers also rely on the assump- determine the optimal configuration of the bubbles. tion that the bubbles in each candidate are connected. As the number of bubbles N increases then so does That is, no bubble may be split into more than one part the complexity of the system and for N ≥ 5 numerical in an attempt to reduce the total perimeter. This is some- methods must be employed. For example, fig. 1 shows the thing that is difficult to prove to be the case in general, two three-connected “simple” partitions of the disc into but which has been shown to be true of all known mini- N = 5 bubbles with equal area. The difference in perime- mizers [23]. ter comes from the different structural arrangements of We use the graph-enumeration software CaGe [24] to the arcs separating the bubbles. If we allow three bub- generate every graph (using the “3-regular Plane Graphs” bles to have one area and the other two a different area generator with its default values), and an associated em- then there are 20 possible structures. When N = 10 this bedding, for each value of N (which corresponds to graphs number increases to 314748. with 2N − 2 vertices). This information is stored as a list We will use combinatorial arguments to enumerate the of vertices, each with an (x, y) position and a list of neigh- graphs corresponding to all possible structures. We recog- bours. The number of graphs for each N is given in table 1. nise that all structures must obey Plateau’s laws [21], a The Surface Evolver [25] is a finite element software consequence of perimeter minimization [22], which state for the minimization of energy subject to constraints. We that edges have constant curvature and meet in threes at convert the CaGe output into a 2D Surface Evolver input an angle of 2π/3. Rather than applying these directly, we file [15], in which each edge is represented as an arc of a will rely on standard numerical minimization software to circle (circular arc mode) and the relevant energy is the determine the equilibrated configuration for each choice sum of edge lengths. The cluster is confined within a cir- of N and areas. cular constraint with unit area, and we set a target area for each bubble. The Evolver’s minimization routines are then used to find a minimum of the perimeter for each 2 Enumeration and evaluation of candidate topology and target areas. structures If an edge shrinks to zero length during the mini- mization, this is not a topology that will give rise to We consider each structure to be a simple, three-regular a stable candidate, since four-fold vertices are not min- (cubic), three-connected planar graph (fig. 2). This allows imizing. We therefore allow topological changes when an is to use graph-generating software to enumerate all the edge shrinks below a critical value lc (we use lc =0.01, relevant partitions of the disc, and then to minimize each which is less than 1/50th of the disc radius). This prevents one’s perimeter to determine the optimum. We therefore time-consuming calculation of non-optimal candidates, hypothesise a one-to-one correspondence between these but does result in some solutions being found repeatedly graphs and the candidate solutions to the least perime- as the result of different topological changes on different ter partition.
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