Optimal Packings for Rings of Circles
- This article is an introduction to rings of circles and their applications.
- You can download a MATLAB app to generate optimal packings of filled rings of circles here.
- See “Optimal Packings for Filled Rings of Circles” for more information.
We define Ring(m,r,s) as the set of m congruent discs with circular boundaries of radius r > 0, whose centers are regularly spaced on a common core circle of radius s centered at a point O, and whose interiors are disjoint; see Figure 1. As indicated in part (b) of Figure 1, the discs may be externally tangent to each other on their boundary circles, in which case we say the ring is filled.
Consider two concentric rings such that the core circle of one ring lies outside of the core circle of the other. We say the two rings are arranged orderly if the interiors of the discs from the two rings are disjoint, but there exists at least one point of tangency between discs from the two rings; see Figure 2.
We say the orderly packing has minimal separation with respect to a fixed initial radius if the packing minimizes successive radii of the rings over all such orderly packings. One may consider minimal separation of rings with either identical or unequal discs in separate rings. Optimal packing of discs in rings with a variety of possible arrangements has an array of applications, some of which are demonstrated below. |
Figure 1. (a) A ring with 5 congruent discs of radius r, with centers regularly spaced on a core circle of radius s centered at O. (b) A filled ring with 9 congruent discs of radius r, where consecutive discs are externally tangent to each other.
Figure 2. (a) An orderly packing of two concentric rings with 7 and 5 congruent discs. (b) An orderly packing of two filled rings with 13 and 9 congruent discs.
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High voltage power cable designs
Aluminum conductor steel-reinforced (ACSR) cable is a high-capacity, high-strength stranded conductor used in overhead power lines. The outer strands are aluminum, chosen for its high conductivity, and the center strands are steel, chosen to increase the strength of the cable. Each strand has a circular cross section. High strand packing density is achieved by placing the wires in rings. Strand conductor rings are often compressed to reduce the diameter. After compression, the aluminum strands no longer have precisely circular cross sections. In practice, it is appropriate to choose a radius that is slightly smaller than the optimal radius for aluminum strands to allow for compression. Figure 3 shows some possible packings with minimal separation useful in cable designs.
Figure 3. Minimally separated rings useful in ACSR cable designs. Possible steel strands are in steel blue and aluminum strands are in light gray. (a) A possible configuration with 7 steel and 18 aluminum strands. The packing density is 0.8050. (b) A configuration with 19 steel and 30 aluminum strands. The packing density is 0.8120. Industrial realization is available with approximately 5/3 aluminum to steel diameter [1]. (c) A possible configuration with 7 steel and 48 aluminum strands. The packing density is 0.7816.
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Circle packing in a circle
Any packing of circles within a circle admits a ring structure if one allows for irregular spacing of circles along each ring. For example, the optimal solution for 13 circles in a circle (proved by Fodor [2]) can be viewed as three unfilled rings of radius \(\sqrt{5}-1\), \(2\), and \(\sqrt{5}+1\), containing 3, 1, and 9 unit circles, respectively; see Figure 4(a). The container radius is \(\sqrt{5}+2\). Since a \(\sqrt{5}+2\) radius container can cover a ring with 10 unit circles, this solution is not unique. Next we consider two conjectured circle packing solutions (see Figure 4(b) and 4(c)). For n=15, the conjectured optimal solution is on 2 rings. For n=17, the conjectured optimal solution is on 6 rings.
For a given number of circles, there is a finite number of possible rings. If we can establish that there are only finitely many optimal configurations, then it will be a significant breakthrough towards establishing the optimality of conjectured computational solutions. In our article, we accomplished this for the special case of filled rings of circles.
For a given number of circles, there is a finite number of possible rings. If we can establish that there are only finitely many optimal configurations, then it will be a significant breakthrough towards establishing the optimality of conjectured computational solutions. In our article, we accomplished this for the special case of filled rings of circles.
Figure 4. (a) The optimal packing for 13 unit circles in a circle consists of three rings of circles. (b) The conjected optimal packing for 15 unit circles in a circle consists of two rings of circles. (c) The conjected optimal packing for 17 unit circles in a circle consists of six rings of circles.
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Visualization
Rings of circles provide a visual representation for some planer symmetry groups. The symmetry group of any set of concentric filled orderly packed rings with minimal separation is a dihedral group of some order. Figure 5 shows four examples of minimal packings with different symmetry groups.
Figure 5. The symmetry group for the minimal packing of two concentric rings is the dihedral group: (a) \(D_1\) (b) \(D_2\) (c) \(D_3\) (d) \(D_4\).
We can also use rings to visualize greatest common divisors. For example, optimally configured filled rings with 8, 12, and 20 circles can be separated into at most 4 equal sections and 4 is the greatest common divisor for these numbers. See Figure 6. Finally, along with some color choices, rings of circles can be visually striking. See Figure 7.
References:
[1] CME Cable and Wire Inc., ACSR Aluminum Conductor, Accessed 9 March (2020), http://www.cmewire.com/catalog/sec03-bac/bac-08-acsrtp.pdf.
[2] F. Fodor, The densest packing of 13 congruent circles in a circle, Beitr. Algebra Geom. 44 (2003), pp. 431--440.
[2] F. Fodor, The densest packing of 13 congruent circles in a circle, Beitr. Algebra Geom. 44 (2003), pp. 431--440.