General circle packings are arrangements of circles on a given surface such that no two circles overlap except at tangent points. In this paper, we examine the optimal arrangement of circles centered on concentric annuli, in what we term rings. Our motivation for this is two-fold: first, certain industrial applications of circle packing naturally allow for filled rings of circles; second, any packing of circles within a circle admits a ring structure if one allows for irregular spacing of circles along each ring. As a result, the optimization problem discussed herein will be extended in a subsequent paper to a more general setting. With this framework in mind, we present properties of concentric rings that have common points of tangency, the exact solution for the optimal arrangement of filled rings along with its symmetry group, and applications to construction of aluminum-conductor steel reinforced cables.
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We dedicate this paper to the memory of Dr. Iraj Kalantari who had great interest in tiling problems. The authors are thankful for many useful discussions they had with Iraj on circle packing. The authors wish to thank the anonymous reviewers for their helpful comments to improve the paper.
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Ekanayake, D.B., Ranpatidewage, M.M. & LaFountain, D.J. Optimal Packings for Filled Rings of Circles. Appl Math 65, 1–22 (2020). https://doi.org/10.21136/AM.2020.0244-19
- minimal separation
- dense packing