Least-perimeter partition of the disc into N regions of two different areas
aa r X i v : . [ c ond - m a t . s o f t ] J a n Least-perimeter partition of the disc into N regions of two differentareas F. J. Headley and S. J. CoxDepartment of Mathematics, Aberystwyth University, SY23 3BZ, UK.December 2018
Abstract
We present conjectured candidates for the least perimeter partition of a disc into N ≤ N . Candidate struc-tures are obtained by assigning different areas to the regions: for even N there are N/ N/ N we consider both cases, i.e. wherethe extra region takes either the larger or the smaller area. The perimeter of each candidate isfound numerically for a few representative area ratios, and then the data is interpolated to givethe conjectured least perimeter candidate for all possible area ratios. At larger N we find thatthese candidates are best for a more limited range of the area ratio. Due to their structural stability and low material cost, energy-minimizing structures have a widearray of applications [1]. In engineering an example is the Beijing Aquatics Centre, which usesslices of the Weaire-Phelan structure [2] to create a lightweight and strong but beautiful piece ofarchitecture.The Weaire-Phelan structure is a solution to the celebrated Kelvin problem, which seeks theminimum surface area partition of space into cells of equal volume [3]. This builds upon thewell-known isoperimetric problem, concerning the least perimeter shape enclosing a given area [4].Extending this idea to many regions with equal areas has led to further rigorous results for optimalstructures, for example the proof of the honeycomb conjecture [5], the optimality of the standardtriple bubble in the plane [6] and of the tetrahedral partition of the surface of the sphere into fourregions [7].If the areas of the regions are allowed to be unequal, then the problem of seeking the configura-tion of least perimeter is more difficult. For N = 2 regions in R , the double bubble conjecture hasbeen proved [8], and, in the plane, the extension of the honeycomb to two different areas (bidis-perse) has led to conjectured solutions [9]. There has also been some experimental work that soughtto correlate the frequency with which different configurations of bidisperse bubble clusters (which,to a good approximation, minimize their surface area [1]) were found with the least perimeterconfiguration [10].Minimal perimeter partitions of domains with a fixed boundary have also generated interest,for example a proof of the optimal partition of the disc into N = 3 regions of given areas [11], and1 a) P =6.304 (b) P = 6 . Figure 1: The two different partitions of the disc into N = 5 regions of equal area. The structureon the right has least perimeter P .many numerical conjectures, e.g. [12, 13, 14, 15]. Such results may lead to further aestheticallypleasing structures like the Water Cube but that are truly foam-like, including their boundary,rather than being unphysical sections through a physical object.In this work we seek to generate and test, in a systematic way, candidate partitions of domainswith fixed boundary. Due to the complexity, and in particular the large number of candidates, werestrict ourselves to a two-dimensional (2D) problem. Thus, we enumerate all partitions of a discand evaluate the perimeter of each one to determine the optimal configuration of the regions.As the number of regions N increases then so does the complexity of the system and for N ≥ N = 5 regions with equal area. The difference in perimetercomes from the different structural arrangements of the arcs separating the regions. If we allow threeregions to have one area and the other two a different area then there are 20 possible structures.When N = 10 this number increases to 314,748.We will use combinatorial arguments to enumerate the graphs corresponding to all possiblestructures. We recognise that all structures must obey Plateau’s laws [16], a consequence of perime-ter minimization [17], which state that edges have constant curvature and meet in threes at an angleof 2 π/
3. Rather than applying these directly, we will rely on standard numerical minimization soft-ware to determine the equilibrated configuration for each choice of N and areas. As the basis for enumerating possible partitions of the disc, we consider each candidate structureas a simple, three-regular (cubic), three-connected planar graph (figure 2). There is a one-to-onecorrespondence between these graphs and the candidate solutions to the least perimeter partition.The assumption of planarity is natural, since these graphs must be embeddable in the 2D disc.The assumption that the graphs are three-regular follows from Plateau’s laws. We assume thatthe graphs are simple and three-connected because any two edges sharing two vertices can bedecomposed into a configuration with lower perimeter P . An example is shown in figure 3: movingthe lens-shaped region to the edge of the disc results in a change in topology and a reduction inperimeter. A similar reduction in perimeter can be achieved in structures with more regions bymoving a lens towards a threefold vertex and performing the same change in topology.2igure 2: A simple cubic three-connected planar graph with three regions of equal area, and itsassociated minimal-perimeter monodisperse partition of the disc.Figure 3: The central two-sided region in this non-simple, two-connected structure (left) can bemoved so that one of its vertices touches the boundary (middle) without changing the perimeter P of this configuration. Once there, a change in topology results in a drop in the perimeter and asimple, three-connected, state (right). 3igure 4: All partitions of the disc into N = 5 regions with area ratio A r = 2 and three large andtwo small regions (5 ). The candidates are shown in order of increasing perimeter (left to right,top row then bottom row). Note how the motif of the two structures in figure 1 is repeated withdifferent arrangements of the two possible areas.We use the graph-enumeration software CaGe [18] to generate every graph and an associatedembedding for each value of N . This information is stored as a list of vertices, each with an ( x, y )position and a list of neighbours. The number of graphs for each N is given in Table 1.The Surface Evolver [19] is finite element software for the minimization of energy subject toconstraints. We convert the CaGe output into a 2D Surface Evolver input file [14], in which eachedge is represented as an arc of a circle and the relevant energy is the sum of edge lengths. Thecluster is confined within a circular constraint with unit area, and we set a target area for eachregion. The Evolver’s minimization routines are then used to find a minimum of the perimeter foreach topology and target areas.If an edge shrinks to zero length during the minimization, this is not a topology that will giverise to a stable candidate, since four-fold vertices are not minimizing. We therefore allow topologicalchanges when an edge shrinks below a critical value l c (we use l c = 0 .
01, which is less than 1/50thof the disc radius). This prevents time-consuming calculation of non-optimal candidates, but doesresult in some solutions being found repeatedly as the result of different topological changes ondifferent candidates.Our aim is to consider bidisperse structures, in which each region can take one of two possibleareas. We define the area ratio A r to be the ratio of the area of the large regions to the area ofthe small regions, so that A r >
1. When the smaller regions are very small, the precise area ratiochanges the total energy only very little, so we consider A r up to 10. (The highest area ratio atwhich we find a change in the topology of the optimal structure is A r = 8 . N is even) or (when N is odd) as close as possible. In the latter case, weconsider both possibilities: one extra large region or one extra small region; see figure 4. We labela configuration with N L large regions and S = N − N L small regions as N LS . For each graph wepermute all possible arrangements of the areas of the N regions (with some redundancy).For example, for N = 3, there is only one possible graph (figure 2), in which three lines meettogether in an internal vertex, as for the monodisperse case. Since N is odd we consider 3 and 3 separately. In the first case there are three possible permutations of the areas assignedto the three regions, but all three are clearly equivalent through a rotation, so there is only onecandidate for which the perimeter must be evaluated. In the second case there are also three4 Graphs Permutations Total Foams A r = 2 A r = 4 A r = 6 A r = 8 A r = 104 1 4 4 4 4 4 3 35 2 10 20 9 7 6 5 69 8 7 8 86 5 20 100 31 25 19 17 197 14 35 490 136 100 74 76 76139 96 78 75 768 50 70 3500 711 495 377 358 3809 233 126 29358 3716 2619 2072 1949 19623608 2562 2074 1958 197110 1249 252 314748 22145 15217 12536 11990 12008Table 1: For each number of regions N we show the number of simple, cubic, three-connectedgraphs, the number of permutations of the two possible areas (for odd N this is half of the numberof structures tested), and then the product, which is the number of candidates whose perimeterwe evaluate. The last five columns give the number of distinct realizable structures found afterminimization, for each area ratio. For odd N the candidates with one extra large region are shownin the top row for each N .possible permutations of the area, but again only one candidate needs to be minimized.The number of graphs and the number of area permutations rises rapidly. We therefore treatonly values of N between 4 and 10. The number of candidates that we evaluate and the numberof structures that are actually realized is shown in Table 1. The perimeter P decreases quite strongly with increasing area ratio, because small enough regionsmake only a small perturbation to a structure with lower N , and structures with lower N havelower P . Although the average area of each region is fixed (at 1 /N ), the polydispersity increaseswith A r . A general measure of polydispersity for regions i with areas A iN is p = q h A iN ih q A iN i − , (1)where h i denotes an average over i . Note that with this definition p = 0 for a monodispersepartition. For a partition with N L large regions this becomes p = q N L N A r + (1 − N L N ) N L N √ A r + (1 − N L N ) − . (2)We expect the perimeter to decrease as 1 / (1+ p ) [20], and so to help distinguish different candidatesfor given N over a range of area ratio A r , we plot P (1 + p ) in the following.Figures 5–11, for N = 4 to 10 respectively, show the scaled perimeter of the structures analysed.The optimal perimeter for each N and each A r is highlighted with a thick line, the transitions5etween structures are indicated, and the least perimeter structures themselves are shown accordingto the area ratio at which they are found.We start by investigating area ratios A r = 2 , , , N = 4 and 5 there is no changein the topology of our conjectured least perimeter structure as the area ratio changes; see figures 5and 6. For N = 4 the two smaller regions never touch, and lie at opposite ends of a straight centraledge. For N = 5, for both possible distributions of large and small regions, the optimal patternalways consists of two three-sided regions whose internal vertices are connected to the other internalvertex, which itself has one other connection to the boundary of the disc. That is, in neither casedoes the optimal candidate have an internal region.For N ≥ A r changes. Wetherefore interpolate between these values of area ratio to determine the critical values of A r atwhich the changes in topology of the least perimeter candidate occur for each N .We do this by taking each of the structures that was found for each area ratio A r = 2 , , . . . and change the area ratio in small steps (of 0.05). For each of these candidates we find and recordthe perimeter. (For N = 10 we do this only for the fifty or so best candidates for each value of A r ,since there are so many candidates which are far from optimal for any area ratio.) For candidateswhose initial area ratio was 2, 4 or 6 we decreased the area ratio to 1.1 and the increased it up to10. For candidates whose initial area ratio was 8 or 10 we increased the area ratio up to 10 beforeslowly decreasing it down to 1.1. We are therefore able to confirm that at low enough area ratiowe recover the optimal structures found in the monodisperse case [21].This procedure generates a few extra optimal structures that are missed by the first samplingof the area ratios, for example between A r = 2 and 4 for N = 8 , and 9 and between A r = 4and 6 for N = 9 and for N = 10.For N = 6 we find that the topology of the least perimeter candidates with area ratios A r ≥ . N = 7 behave differently (figure 8). In the case 7 there are onlytwo different candidates found, and for A r ≥ . we find four different topologies, with a transition to a new candidate at a surprisingly higharea ratio of 8.4.The least perimeter structure with N = 8 regions has the monodisperse topology for A r < . A r = 3 .
9, giving three different optimal structures.For N = 9 the results are richer (figure 10), in the sense that the system explores more possiblestates as the area ratio changes. For 9 we find five different topologies, while for 9 there arefour. In the latter case the structure found for A r = 2 is different to the monodisperse one [21],and there is a transition to that structure at a low area ratio around 1.8.Finally, for N = 10 (figure 11) we again find a candidate for A r = 2 that differs from themonodisperse case and a transition at even lower area ratio. In total there are five differenttopologies. The critical area ratios at which there is a transition between optimal structures are summarisedin figure 12. Most are found at intermediate values of the area ratio, roughly between A r = 2 . N . There is also a single point at high A r , for N = 7, which corresponds to moving a small bubble from the boundary of the disc to the centre,6 S c a l ed P e r i m e t e r P ( + p ) Area ratio
Figure 5: The perimeter P of the least perimeter candidates N = 4 at different area ratios.(a) S c a l ed P e r i m e t e r P ( + p ) Area ratio (b) S c a l ed P e r i m e t e r P ( + p ) Area ratio
Figure 6: The perimeter P of the least perimeter candidates N = 5 at different area ratios, for (a)the case with one extra large region 5 and (b) one extra small region 5 . Sudden drops in P correspond to topological changes when an edge shrinks to zero length.7 S c a l ed P e r i m e t e r P ( + p ) Area ratio
Figure 7: The perimeter P of the least perimeter candidates N = 6 at different area ratios. Thetransition between the two optimal structures are marked by a black dot.(a) S c a l ed P e r i m e t e r P ( + p ) Area ratio (b) S c a l ed P e r i m e t e r P ( + p ) Area ratio
Figure 8: The perimeter P of the least perimeter candidates N = 7 at different area ratios, for (a)the case with one extra large region 7 and (b) one extra small region 7 .8 S c a l ed P e r i m e t e r P ( + p ) Area ratio
Figure 9: The perimeter P of the least perimeter candidates N = 8 at different area ratios. Weonly show the perimeter corresponding to the fifty best candidates for each area ratio.(a) S c a l ed P e r i m e t e r P ( + p ) Area ratio (b) S c a l ed P e r i m e t e r P ( + p ) Area ratio
Figure 10: The perimeter P of the least perimeter candidates N = 9 at different area ratios, for(a) the case with one extra large region 9 and (b) one extra small region 9 . We only show theperimeter corresponding to the fifty best candidates for each area ratio.9 S c a l ed P e r i m e t e r P ( + p ) Area ratio
Figure 11: The perimeter P of the least perimeter candidates N = 10 at different area ratios. Weonly show the perimeter corresponding to the fifty best candidates for each area ratio. C r i t i c a l a r ea r a t i o Number of regions
Figure 12: The area ratio at which there is a transition between different least perimeter arrange-ments. For odd N , diamonds refer to the case with one extra large region.and hence to a symmetric state. It is perhaps surprising that this highly symmetric state is notoptimal at lower area ratio, since many of the least perimeter structures are symmetric.The images in figures 5–11 also hint at an evolution from the small regions clustering togetherat low area ratio to being separated from each other by the large regions at high area ratio.We quantify this observation by counting the proportion of edges E LS separating large from smallregions in each least perimeter structure. A structure with a higher value of E LS has less clustering.The data in figure 13 bears out this observation: for N ≥ E LS islower than for large area ratio. (The exception is one of the structures for N = 10, where even atlow area ratio (4 . ≤ A r ≤ .
2) the least perimeter candidate has the small bubbles well-separated.)
We have enumerated all candidate partitions of the disc with N ≤
10 regions with one of twodifferent areas, and determined, for each area ratio, the partition with least perimeter. The resultsshow an increasing number of transitions between the different optimal structures found for varyingarea ratio as N increases, mostly at low area ratio. Further, in the least perimeter partitions atsmall area ratio the smaller regions are clustered together, while at large area ratio the small10 P r opo r t i on o f s m a ll - l a r ge edge s Number of regionsArea ratio less than 5Area ratio greater than 5
Figure 13: The proportion of edges separating large from small regions E ls in the least perimetercandidate for each N . The area ratio is distinguished by whether it is greater or less than five.Data for the cases of an extra large or an extra small region is collated.regions are separated by large regions. Transitions between such mixed and sorted configurationsoften occur as a consequence of some agitation [22].The procedure described here should translate directly to least perimeter partitions of thesurface of a sphere, since to enumerate candidates to that problem we are able to use the samegraphs and consider the periphery of the graph to form the boundary of one further region. Thus thecandidates for the disc with N regions are also the candidates for the sphere with N + 1 regions. Ingeneral, our preliminary results indicate, as for the monodisperse case [14], that the least perimeterarrangement of regions on the sphere is different to the corresponding optimal partition of the disc. Acknowledgements
We thank DG Evans for helpful discussions. FJH was supported by a Walter Idris Jones ResearchScholarship and SJC by the UK Engineering and Physical Sciences Research Council (EP/N002326/1).
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