Least-perimeter partition of the disc into $N$ regions of two different areas

TL;DR

This paper conjectures optimal partitions of a disk into N regions of two areas, using graph enumeration and numerical methods to identify least-perimeter configurations for N up to 10.

Contribution

It introduces a method to identify candidate least-perimeter partitions for two-area regions in a disk by enumerating three-connected cubic graphs and interpolating perimeter data.

Findings

Candidates are identified for N ≤ 10 regions.

Optimal configurations depend on the area ratio.

Candidates are best for specific area ratios at larger N.

Abstract

We present conjectured candidates for the least perimeter partition of a disc into $N \le 10$ regions which take one of two possible areas. We assume that the optimal partition is connected, and therefore 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 $N/2$ regions of one area and $N/2$ regions of the other, and for odd $N$ we consider both cases, i.e. where the extra region takes either the larger or the smaller area. The perimeter of each candidate 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. At larger $N$ we find that these candidates are best for a more limited range of the area ratio.

Tables

Table 1. Table 1: For each number of regions N 𝑁 N we show the number of simple, cubic, three-connected graphs, the number of permutations of the two possible areas (for odd N 𝑁 N this is half of the number of structures tested), and then the product, which is the number of candidates whose perimeter we evaluate. The last five columns give the number of distinct realizable structures found after minimization, for each area ratio. For odd N 𝑁 N the candidates with one extra large region are shown in the top row for each N 𝑁 N .

$N$	Graphs	Permutations	Total Foams	$A_r=2$	$A_r=4$	$A_r=6$	$A_r=8$	$A_r=10$
4	1	4	4	4	4	4	3	3
5	2	10	20	9	7	6	5	6
				9	8	7	8	8
6	5	20	100	31	25	19	17	19
7	14	35	490	136	100	74	76	76
				139	96	78	75	76
8	50	70	3500	711	495	377	358	380
9	233	126	29358	3716	2619	2072	1949	1962
				3608	2562	2074	1958	1971
10	1249	252	314748	22145	15217	12536	11990	12008

Equations

$$p=\frac{\sqrt{⟨A_N^i⟩}}{⟨\sqrt{A_N^i}⟩}-1,$$

$$p=\frac{\sqrt{\frac{N_L}{N}{A}_r+(1-\frac{N_L}{N})}}{\frac{N_L}{N}\sqrt{A_r}+(1-\frac{N_L}{N})}-1.$$

Full text

1 Introduction

Due to their structural stability and low material cost, energy-minimizing structures have a wide array of applications [1]. In engineering an example is the Beijing Aquatics Centre, which uses slices of the Weaire-Phelan structure [2] to create a lightweight and strong but beautiful piece of architecture.

The Weaire-Phelan structure is a solution to the celebrated Kelvin problem, which seeks the minimum surface area partition of space into cells of equal volume [3]. This builds upon the well-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 optimal structures, for example the proof of the honeycomb conjecture [5], the optimality of the standard triple bubble in the plane [6] and of the tetrahedral partition of the surface of the sphere into four regions [7].

If the areas of the regions are allowed to be unequal, then the problem of seeking the configuration of least perimeter is more difficult. For $N=2$ regions in ${{R}}^{3}$, the double bubble conjecture has been proved [8], and, in the plane, the extension of the honeycomb to two different areas (bidisperse) has led to conjectured solutions [9]. There has also been some experimental work that sought to 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 perimeter configuration [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], and many numerical conjectures, e.g. [12, 13, 14, 15]. Such results may lead to further aesthetically pleasing 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 domains with fixed boundary. Due to the complexity, and in particular the large number of candidates, we restrict ourselves to a two-dimensional (2D) problem. Thus, we enumerate all partitions of a disc and 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 $N\geq 5$ numerical methods must be employed. For example, figure 1 shows the two three-connected “simple” partitions of the disc into $N=5$ regions with equal area. The difference in perimeter comes from the different structural arrangements of the arcs separating the regions. If we allow three regions 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 possible structures. We recognise that all structures must obey Plateau’s laws [16], a consequence of perimeter minimization [17], which state that edges have constant curvature and meet in threes at an angle of $2\pi/3$. Rather than applying these directly, we will rely on standard numerical minimization software to determine the equilibrated configuration for each choice of $N$ and areas.

2 Enumeration and evaluation of candidate structures

As the basis for enumerating possible partitions of the disc, we consider each candidate structure as a simple, three-regular (cubic), three-connected planar graph (figure 2). There is a one-to-one correspondence 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 that the graphs are simple and three-connected because any two edges sharing two vertices can be decomposed into a configuration with lower perimeter $P$. An example is shown in figure 3: moving the lens-shaped region to the edge of the disc results in a change in topology and a reduction in perimeter. A similar reduction in perimeter can be achieved in structures with more regions by moving a lens towards a threefold vertex and performing the same change in topology.

We use the graph-enumeration software CaGe [18] to generate every graph and an associated embedding 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 to constraints. We convert the CaGe output into a 2D Surface Evolver input file [14], in which each edge is represented as an arc of a circle and the relevant energy is the sum of edge lengths. The cluster is confined within a circular constraint with unit area, and we set a target area for each region. The Evolver’s minimization routines are then used to find a minimum of the perimeter for each topology and target areas.

If an edge shrinks to zero length during the minimization, this is not a topology that will give rise to a stable candidate, since four-fold vertices are not minimizing. We therefore allow topological changes when an edge shrinks below a critical value $l_{c}$ (we use $l_{c}=0.01$, which is less than 1/50th of the disc radius). This prevents time-consuming calculation of non-optimal candidates, but does result in some solutions being found repeatedly as the result of different topological changes on different candidates.

Our aim is to consider bidisperse structures, in which each region can take one of two possible areas. We define the area ratio $A_{r}$ to be the ratio of the area of the large regions to the area of the small regions, so that $A_{r}>1$. When the smaller regions are very small, the precise area ratio changes the total energy only very little, so we consider $A_{r}$ up to 10. (The highest area ratio at which we find a change in the topology of the optimal structure is $A_{r}=8.35$.)

To reduce the number of possible candidates, we stipulate that the number of regions of each area are equal (when $N$ is even) or (when $N$ is odd) as close as possible. In the latter case, we consider both possibilities: one extra large region or one extra small region; see figure 4. We label a configuration with $N_{L}$ large regions and $S=N-N_{L}$ small regions as $N_{LS}$. For each graph we permute 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 meet together in an internal vertex, as for the monodisperse case. Since $N$ is odd we consider $3_{21}$ and $3_{12}$ separately. In the first case there are three possible permutations of the areas assigned to the three regions, but all three are clearly equivalent through a rotation, so there is only one candidate for which the perimeter must be evaluated. In the second case there are also three 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 treat only values of $N$ between 4 and 10. The number of candidates that we evaluate and the number of structures that are actually realized is shown in Table 1.

3 Results

3.1 Least perimeter candidates at representative area ratios

The perimeter $P$ decreases quite strongly with increasing area ratio, because small enough regions make only a small perturbation to a structure with lower $N$, and structures with lower $N$ have lower $P$. Although the average area of each region is fixed (at $1/N$), the polydispersity increases with $A_{r}$. A general measure of polydispersity for regions $i$ with areas $A_{N}^{i}$ is

[TABLE]

where $\langle\;\rangle$ denotes an average over $i$. Note that with this definition $p=0$ for a monodisperse partition. For a partition with $N_{L}$ large regions this becomes

[TABLE]

We expect the perimeter to decrease as $1/(1+p)$ [20], and so to help distinguish different candidates for 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 transitions between structures are indicated, and the least perimeter structures themselves are shown according to the area ratio at which they are found.

We start by investigating area ratios $A_{r}=2,4,6,8$ and $10$. For $N=4$ and $5$ there is no change in the topology of our conjectured least perimeter structure as the area ratio changes; see figures 5 and 6. For $N=4$ the two smaller regions never touch, and lie at opposite ends of a straight central edge. For $N=5$, for both possible distributions of large and small regions, the optimal pattern always consists of two three-sided regions whose internal vertices are connected to the other internal vertex, which itself has one other connection to the boundary of the disc. That is, in neither case does the optimal candidate have an internal region.

For $N\geq 6$ there are transitions between different structures as the area ratio $A_{r}$ changes. We therefore interpolate between these values of area ratio to determine the critical values of $A_{r}$ at which 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 $A_{r}=2,4,\ldots$ and change the area ratio in small steps (of 0.05). For each of these candidates we find and record the 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 candidates whose initial area ratio was 2, 4 or 6 we decreased the area ratio to 1.1 and the increased it up to 10. For candidates whose initial area ratio was 8 or 10 we increased the area ratio up to 10 before slowly decreasing it down to 1.1. We are therefore able to confirm that at low enough area ratio we recover the optimal structures found in the monodisperse case [21].

This procedure generates a few extra optimal structures that are missed by the first sampling of the area ratios, for example between $A_{r}=2$ and 4 for $N=8,9_{54}$ and $9_{45}$ and between $A_{r}=4$ and $6$ for $N=9_{54}$ and for $N=10$.

For $N=6$ we find that the topology of the least perimeter candidates with area ratios $A_{r}\geq 2.6$ are the same (figure 7). For area ratios less than this value the topology is that of the optimal candidate in the monodisperse case [21].

The two different cases for $N=7$ behave differently (figure 8). In the case $7_{43}$ there are only two different candidates found, and for $A_{r}\geq 2.8$ the topology does not change. On the other hand, for $7_{34}$ we find four different topologies, with a transition to a new candidate at a surprisingly high area ratio of 8.4.

The least perimeter structure with $N=8$ regions has the monodisperse topology for $A_{r}<2.6$ (figure 9); there is one further transition at $A_{r}=3.9$, giving three different optimal structures.

For $N=9$ the results are richer (figure 10), in the sense that the system explores more possible states as the area ratio changes. For $9_{54}$ we find five different topologies, while for $9_{45}$ there are four. 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 the monodisperse case and a transition at even lower area ratio. In total there are five different topologies.

3.2 Analysis of patterns

The critical area ratios at which there is a transition between optimal structures are summarised in figure 12. Most are found at intermediate values of the area ratio, roughly between $A_{r}=2.5$ and 4, although this broadens slightly with increasing $N$. 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, and hence to a symmetric state. It is perhaps surprising that this highly symmetric state is not optimal 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 together at 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 small regions 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 $N\geq 6$ and small area ratio the value of $E_{LS}$ is lower than for large area ratio. (The exception is one of the structures for $N=10$, where even at low area ratio ($4.5\leq A_{r}\leq 5.2$) the least perimeter candidate has the small bubbles well-separated.)

4 Conclusions

We have enumerated all candidate partitions of the disc with $N\leq 10$ regions with one of two different areas, and determined, for each area ratio, the partition with least perimeter. The results show an increasing number of transitions between the different optimal structures found for varying area ratio as $N$ increases, mostly at low area ratio. Further, in the least perimeter partitions at small area ratio the smaller regions are clustered together, while at large area ratio the small regions are separated by large regions. Transitions between such mixed and sorted configurations often occur as a consequence of some agitation [22].

The procedure described here should translate directly to least perimeter partitions of the surface of a sphere, since to enumerate candidates to that problem we are able to use the same graphs and consider the periphery of the graph to form the boundary of one further region. Thus the candidates for the disc with $N$ regions are also the candidates for the sphere with $N+1$ regions. In general, our preliminary results indicate, as for the monodisperse case [14], that the least perimeter arrangement 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 Research Scholarship and SJC by the UK Engineering and Physical Sciences Research Council (EP/N002326/1).