{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T03:12:45Z","timestamp":1785553965403,"version":"3.56.0"},"reference-count":40,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,9]],"date-time":"2021-11-09T00:00:00Z","timestamp":1636416000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012550","name":"Nemzeti Kutat\u00e1si, Fejleszt\u00e9si \u00e9s Innovaci\u00f3s Alap","doi-asserted-by":"publisher","award":["K119440"],"award-info":[{"award-number":["K119440"]}],"id":[{"id":"10.13039\/501100012550","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003549","name":"Hungarian Scientific Research Fund","doi-asserted-by":"publisher","award":["K81146"],"award-info":[{"award-number":["K81146"]}],"id":[{"id":"10.13039\/501100003549","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by n congruent circles of given radius r, while r varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for n = 2 to 5. In the present paper, numerical solutions are given to this problem for n = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of n = 6 is very complicated, whilst the case n = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.<\/jats:p>","DOI":"10.3390\/sym13112133","type":"journal-article","created":{"date-parts":[[2021,11,9]],"date-time":"2021-11-09T21:39:07Z","timestamp":1636493947000},"page":"2133","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Partial Covering of a Circle by 6 and 7 Congruent Circles"],"prefix":"10.3390","volume":"13","author":[{"given":"Zsolt","family":"G\u00e1sp\u00e1r","sequence":"first","affiliation":[{"name":"Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tibor","family":"Tarnai","sequence":"additional","affiliation":[{"name":"Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kriszti\u00e1n","family":"Hincz","sequence":"additional","affiliation":[{"name":"Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"181","DOI":"10.6028\/jres.066B.020","article-title":"Black box maximization of circular coverage","volume":"66","author":"Zahn","year":"1962","journal-title":"J. Res. Nat. Bur. Stand. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"S328","DOI":"10.1097\/00003086-200110001-00030","article-title":"Mosaicplasty for the treatment of articular defects of the knee and ankle","volume":"391","author":"Hangody","year":"2001","journal-title":"Clic. Orthop. Relat. Res."},{"key":"ref_3","first-page":"192","article-title":"Sets of points with given minimum separations (Solutions to Problem E 1921)","volume":"75","author":"Graham","year":"1968","journal-title":"Amer. Math. Mon."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1002\/mana.19690400110","article-title":"Der Mindestabstand von n in der Einheitskreisscheibe gelagenen Punkten","volume":"40","author":"Pirl","year":"1969","journal-title":"Math. Nachr."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF01263647","article-title":"Densest packing of eleven congruent circles in a circle","volume":"50","author":"Melissen","year":"1994","journal-title":"Geom. Dedicata."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1023\/A:1005091317243","article-title":"The densest packing of 19 congruent circles in a circle","volume":"74","author":"Fodor","year":"1999","journal-title":"Geom. Dedicata."},{"key":"ref_7","first-page":"401","article-title":"The densest packing of 12 congruent circles in a circle","volume":"41","author":"Fodor","year":"2000","journal-title":"Beitr\u00e4ge Algebra Geom."},{"key":"ref_8","first-page":"431","article-title":"The densest packing of 13 congruent circles in a circle","volume":"44","author":"Fodor","year":"2003","journal-title":"Beitr\u00e4ge Algebra Geom."},{"key":"ref_9","first-page":"25","article-title":"Packing of 14 congruent circles in a circle","volume":"16","author":"Fodor","year":"2003","journal-title":"Stud. Univ. \u017dilina Math. Ser."},{"key":"ref_10","unstructured":"Bezdek, K. (1979). Optimal Covering of Circles, Thesis. (In Hungarian)."},{"key":"ref_11","first-page":"7","article-title":"\u00dcber einige Kreis\u00fcberdeckungen","volume":"14","author":"Bezdek","year":"1983","journal-title":"Beitr\u00e4ge Algebra Geom."},{"key":"ref_12","first-page":"361","article-title":"Thinnest covering of a circle by eight, nine, or ten congruent circles","volume":"52","author":"Goodman","year":"2005","journal-title":"Combinatorial and Computational Geometry"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/S0012-365X(97)00050-2","article-title":"Dense packings of congruent circles in a circle","volume":"181","author":"Graham","year":"1998","journal-title":"Discrete Math."},{"key":"ref_14","unstructured":"Nurmela, K.J. (1998). Covering a Circle by Congruent Circular Discs, Department of Computer Science and Engineering, Helsinki University of Technology. preprint."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1515\/advgeom-2016-0018","article-title":"Optimal cover of a disk with three smaller congruent disks","volume":"16","author":"Szalkai","year":"2016","journal-title":"Adv. Geom."},{"key":"ref_16","first-page":"137","article-title":"Partial covering of the unit circle by four equal circles","volume":"60","author":"Tarnai","year":"2017","journal-title":"Annales Univ. Sci. Budapest."},{"key":"ref_17","first-page":"126","article-title":"Partial covering of a circle by equal circles, Part II: The case of 5 circles","volume":"5","author":"Tarnai","year":"2014","journal-title":"J. Comput. Geom."},{"key":"ref_18","unstructured":"Coxeter, H.S.M. (1961). Introduction to Geometry, Wiley."},{"key":"ref_19","unstructured":"Weil, H. (1952). Symmetry, Princeton University Press."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Poston, T., and Stewart, I. (1978). Catastrophe Theory and Its Applications, Pitman.","DOI":"10.1063\/1.2995174"},{"key":"ref_21","first-page":"66","article-title":"\u00dcber die Abschetzung des k\u00fcrzesten Abstandes zweier Punkte eines auf einer Kugelfl\u00e4che liegenden Punktsystems","volume":"53","year":"1943","journal-title":"Jber. Dtsch. Math.-Ver."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1007\/BF02054944","article-title":"Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand ein Platz?","volume":"123","year":"1951","journal-title":"Math. Annln."},{"key":"ref_23","unstructured":"Danzer, L. (1963). Endliche Punktmengen auf der 2-Sph\u00e4re mit m\u00f6glichst grossem Minimalabstand, Habilitationschrift: Universit\u00e4t G\u00f6ttingen."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1007\/s10958-014-2174-7","article-title":"Enumeration of irreducible contact graphs on the sphere","volume":"203","author":"Musin","year":"2014","journal-title":"J. Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1080\/10586458.2015.1022842","article-title":"The Tammes problem for N = 14","volume":"24","author":"Musin","year":"2015","journal-title":"Experimental Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF01396539","article-title":"Arrangement of 24 points on a sphere","volume":"144","author":"Robinson","year":"1961","journal-title":"Math. Annln."},{"key":"ref_27","first-page":"40","article-title":"On covering the spherical surface with equal spherical caps","volume":"50","year":"1943","journal-title":"Matematikai Fiz. Lapok"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/BF01362364","article-title":"\u00dcberdeckung der Kugel mit h\u00f6chstens acht Kreisen","volume":"129","year":"1955","journal-title":"Math. Annln"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1007\/s00454-017-9891-x","article-title":"Covering the sphere with equal circles","volume":"57","author":"Wimmer","year":"2017","journal-title":"Discrete Comp. Geom."},{"key":"ref_30","first-page":"225","article-title":"Kreis\u00fcberdeckungen der Sph\u00e4re","volume":"4","year":"1969","journal-title":"Studia Sci. Math. Hungar."},{"key":"ref_31","unstructured":"Sloane, N.J.A., Hardin, R.H., and Smith, W.D. (2021, October 07). Spherical Codes. Available online: NeilSloane.com\/packings\/."},{"key":"ref_32","unstructured":"Hardin, R.H., Sloane, N.J.A., and Smith, W.D. (2021, October 07). Spherical Coverings. Available online: NeilSloane.com\/coverings\/."},{"key":"ref_33","unstructured":"Fejes T\u00f3th, L. (1964). Regular Figures, Pergamon; Macmillan."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF02095633","article-title":"Perfect distribution of points on a sphere","volume":"1","year":"1971","journal-title":"Periodica Math. Hungar."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2043","DOI":"10.1098\/rspa.1996.0108","article-title":"Transition from spherical circle packing to covering: Geometrical analogues of chemical isomerization","volume":"452","author":"Fowler","year":"1996","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4131","DOI":"10.1098\/rspa.1999.0494","article-title":"Transition from circle packing to covering on a sphere: The odd case of 13 circles","volume":"455","author":"Fowler","year":"1999","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_37","first-page":"104","article-title":"Partial covering of a circle by equal circles, Part I: The mechanical models","volume":"5","author":"Tarnai","year":"2014","journal-title":"J. Comput. Geom."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/PL00009395","article-title":"On the volume of the union of balls","volume":"20","year":"1998","journal-title":"Discrete Comput. Geom."},{"key":"ref_39","unstructured":"Connelly, R. (2008). Maximizing the Area of Unions and Intersections of Discs. Lecture at the Discrete and Convex Geometry Workshop, Alfr\u00e9d R\u00e9nyi Institute of Mathematics."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Pacioli, L. (1494). Summa de Arithmetica, Geometria, Proportioni et Proportionalita, Paganius de Paganinis.","DOI":"10.5479\/sil.440357.39088007406663"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2133\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:28:03Z","timestamp":1760167683000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2133"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,9]]},"references-count":40,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112133"],"URL":"https:\/\/doi.org\/10.3390\/sym13112133","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,9]]}}}