Applications of Mathematics

, Volume 65, Issue 1, pp 1–22 | Cite as

Optimal Packings for Filled Rings of Circles

  • Dinesh B. EkanayakeEmail author
  • Manjula Mahesh Ranpatidewage
  • Douglas J. LaFountain


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.


optimization minimal separation dense packing 

MSC 2010

52C15 52C26 


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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.


  1. [1]
    CME Cable and Wire Inc.: AcuTechTM ACSR, Aluminum Conductor, Steel Reinforced, Twisted Pair Conductors. 2019. Available at
  2. [2]
    F. Fodor: The densest packing of 19 congruent circles in a circle. Geom. Dedicata 74 (1999), 139–145.MathSciNetCrossRefGoogle Scholar
  3. [3]
    R. L. Graham, B. D. Lubachevsky, K. J. Nurmela, P. R. J. Östergård: Dense packings of congruent circles in a circle. Discrete Math. 181 (1998), 139–154.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Y. Li, S. Xu, H. Yang: Design of circular signal constellations in the presence of phase noise. 4th International Conference on Wireless Communications, Networking and Mobile Computing. IEEE, New York, 2008, pp. 2079–2086.Google Scholar
  5. [5]
    C. O. López, J. E. Beasley: Packing a fixed number of identical circles in a circular container with circular prohibited areas. Optim. Lett. 13 (2019), 1449–1468.MathSciNetCrossRefGoogle Scholar
  6. [6]
    D. G. Luenberger, Y. Ye: Linear and Nonlinear Programming. International Series in Operations Research & Management Science 228, Springer, Cham, 2016.CrossRefGoogle Scholar
  7. [7]
    B. G. Mobasseri: Digital modulation classification using constellation shape. Signal Process. 80 (2000), 251–277.CrossRefGoogle Scholar
  8. [8]
    J. P. Pedroso, S. Cunha, J. N. Tavares: Recursive circle packing problems. Int. Trans. Oper. Res. 23 (2016), 355–368.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Y. Stoyan, G. Yaskov: Packing equal circles into a circle with circular prohibited areas. Int. J. Comput. Math. 89 (2012), 1355–1369.MathSciNetCrossRefGoogle Scholar
  10. [10]
    F. R. Thrash, Jr.: Transmission Conductors—A review of the design and selection criteria. Available at (2019), 11 pages.
  11. [11]
    T. Worzyk: Submarine Power Cables. Design, Installation, Repair, Environmental Aspects. Springer, Berlin, 2009.CrossRefGoogle Scholar
  12. [12]
    G. Zoutendijk: Methods of Feasible Directions. A Study in Linear and Non-Linear Programming. Elsevier, Amsterdam, 1960.zbMATHGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Cz 2020

Authors and Affiliations

  • Dinesh B. Ekanayake
    • 1
    Email author
  • Manjula Mahesh Ranpatidewage
    • 1
  • Douglas J. LaFountain
    • 1
  1. 1.Western Illinois UniversityMacomb, ILUSA

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