{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T07:03:22Z","timestamp":1769238202998,"version":"3.49.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,9,14]],"date-time":"2020-09-14T00:00:00Z","timestamp":1600041600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,9,14]],"date-time":"2020-09-14T00:00:00Z","timestamp":1600041600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-1564473"],"award-info":[{"award-number":["DMS-1564473"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS- 1564493"],"award-info":[{"award-number":["DMS- 1564493"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2021,12]]},"DOI":"10.1007\/s00454-020-00242-8","type":"journal-article","created":{"date-parts":[[2020,9,14]],"date-time":"2020-09-14T15:03:46Z","timestamp":1600095826000},"page":"1262-1285","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Packing Disks by Flipping and Flowing"],"prefix":"10.1007","volume":"66","author":[{"given":"Robert","family":"Connelly","sequence":"first","affiliation":[]},{"given":"Steven J.","family":"Gortler","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,9,14]]},"reference":[{"key":"242_CR1","volume-title":"Convex Polyhedra. Springer Monographs in Mathematics","author":"AD Alexandrov","year":"2005","unstructured":"Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005)"},{"issue":"2","key":"242_CR2","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1070\/SM1970v012n02ABEH000920","volume":"12","author":"EM Andreev","year":"1970","unstructured":"Andreev, E.M.: On convex polyhedra of finite volume in Lobachevskii space. Math. USSR-Sbornik 12(2), 255\u2013259 (1970)","journal-title":"Math. USSR-Sbornik"},{"issue":"2","key":"242_CR3","doi-asserted-by":"publisher","first-page":"659","DOI":"10.1090\/S0002-9947-03-03239-2","volume":"356","author":"AI Bobenko","year":"2004","unstructured":"Bobenko, A.I., Springborn, B.A.: Variational principles for circle patterns and Koebe\u2019s theorem. Trans. Am. Math. Soc. 356(2), 659\u2013689 (2004)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1","key":"242_CR4","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1016\/j.comgeo.2008.04.001","volume":"42","author":"P Bose","year":"2009","unstructured":"Bose, P., Hurtado, F.: Flips in planar graphs. Comput. Geom. 42(1), 60\u201380 (2009)","journal-title":"Comput. Geom."},{"issue":"1","key":"242_CR5","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/s00454-017-9928-1","volume":"60","author":"JC Bowers","year":"2018","unstructured":"Bowers, J.C., Bowers, P.L.: Ma\u2013Schlenker $$c$$-octahedra in the $$2$$-sphere. Discrete Comput. Geom. 60(1), 9\u201326 (2018)","journal-title":"Discrete Comput. Geom."},{"key":"242_CR6","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s10711-019-00438-0","volume":"203","author":"JC Bowers","year":"2019","unstructured":"Bowers, J.C., Bowers, P.L., Pratt, K.: Almost all circle polyhedra are rigid. Geom. Dedicata 203, 337\u2013346 (2019)","journal-title":"Geom. Dedicata"},{"issue":"1","key":"242_CR7","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1112\/plms\/s3-73.1.185","volume":"73","author":"P Bowers","year":"1996","unstructured":"Bowers, P., Stephenson, K.: A branched Andreev\u2013Thurston theorem for circle packings of the sphere. Proc. Lond. Math. Soc. 73(1), 185\u2013215 (1996)","journal-title":"Proc. Lond. Math. Soc."},{"issue":"1","key":"242_CR8","doi-asserted-by":"publisher","first-page":"97","DOI":"10.4310\/jdg\/1080835659","volume":"63","author":"B Chow","year":"2003","unstructured":"Chow, B., Luo, F.: Combinatorial Ricci flows on surfaces. J. Differ. Geom. 63(1), 97\u2013129 (2003)","journal-title":"J. Differ. Geom."},{"key":"242_CR9","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1007\/BF01245096","volume":"104","author":"Y Colin de Verdi\u00e8re","year":"1991","unstructured":"Colin de Verdi\u00e8re, Y.: Un principe variationnel pour les empilements de cercles. Invent. Math. 104, 655\u2013669 (1991)","journal-title":"Invent. Math."},{"key":"242_CR10","doi-asserted-by":"crossref","unstructured":"Collins, Ch.R., Stephenson, K.: A circle packing algorithm. Comput. Geom. 25(3), 233\u2013256 (2003)","DOI":"10.1016\/S0925-7721(02)00099-8"},{"issue":"1","key":"242_CR11","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BF01404753","volume":"66","author":"R Connelly","year":"1982","unstructured":"Connelly, R.: Rigidity and energy. Invent. Math. 66(1), 11\u201333 (1982)","journal-title":"Invent. Math."},{"issue":"1","key":"242_CR12","first-page":"78","volume":"60","author":"R Connelly","year":"2013","unstructured":"Connelly, R.: What is $$\\ldots $$ a tensegrity? Not. Am. Math. Soc. 60(1), 78\u201380 (2013)","journal-title":"Not. Am. Math. Soc."},{"key":"242_CR13","doi-asserted-by":"crossref","unstructured":"Connelly, R., Gortler, S.J., Theran, L.: Rigidity for sticky discs. Proc. R. Soc. A 475(2222), #\u00a020180773 (2019)","DOI":"10.1098\/rspa.2018.0773"},{"key":"242_CR14","volume-title":"Regular Figures","author":"L Fejes T\u00f3th","year":"1964","unstructured":"Fejes T\u00f3th, L.: Regular Figures. A Pergamon Press Book, Macmillan, New York (1964)"},{"key":"242_CR15","doi-asserted-by":"crossref","unstructured":"Gluck, H.: Almost all simply connected closed surfaces are rigid. In: Geometric Topology (Park City 1974). Lecture Notes in Math., vol. 438, pp. 225\u2013239. Springer, Berlin (1975)","DOI":"10.1007\/BFb0066118"},{"issue":"9","key":"242_CR16","doi-asserted-by":"publisher","first-page":"4757","DOI":"10.1090\/S0002-9947-2011-05239-6","volume":"363","author":"R Guo","year":"2011","unstructured":"Guo, R.: Local rigidity of inversive distance circle packing. Trans. Am. Math. Soc. 363(9), 4757\u20134776 (2011)","journal-title":"Trans. Am. Math. Soc."},{"issue":"12","key":"242_CR17","doi-asserted-by":"publisher","first-page":"6517","DOI":"10.1090\/S0002-9947-2013-05892-8","volume":"365","author":"Z He","year":"2013","unstructured":"He, Z., Liu, J.: On the Teichm\u00fcller theory of circle patterns. Trans. Am. Math. Soc. 365(12), 6517\u20136541 (2013)","journal-title":"Trans. Am. Math. Soc."},{"key":"242_CR18","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/s10711-008-9339-9","volume":"140","author":"I Izmestiev","year":"2009","unstructured":"Izmestiev, I.: Projective background of the infinitesimal rigidity of frameworks. Geom. Dedicata 140, 183\u2013203 (2009)","journal-title":"Geom. Dedicata"},{"key":"242_CR19","first-page":"141","volume":"88","author":"P Koebe","year":"1936","unstructured":"Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte \u00fcber die Verhandlungen der S\u00e4chsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Kl. 88, 141\u2013164 (1936)","journal-title":"Berichte \u00fcber die Verhandlungen der S\u00e4chsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Kl."},{"key":"242_CR20","doi-asserted-by":"crossref","unstructured":"Lam, W.Y.: Minimal surfaces from infinitesimal deformations of circle packings. Adv. Math. 362, #\u00a0106939 (2020)","DOI":"10.1016\/j.aim.2019.106939"},{"issue":"2","key":"242_CR21","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/0020-0190(90)90142-K","volume":"34","author":"J-P Laumond","year":"1990","unstructured":"Laumond, J.-P.: Connectivity of plane triangulations. Inform. Process. Lett. 34(2), 87\u201396 (1990)","journal-title":"Inform. Process. Lett."},{"issue":"4","key":"242_CR22","doi-asserted-by":"publisher","first-page":"2299","DOI":"10.2140\/gt.2011.15.2299","volume":"15","author":"F Luo","year":"2011","unstructured":"Luo, F.: Rigidity of polyhedral surfaces, III. Geom. Topol. 15(4), 2299\u20132319 (2011)","journal-title":"Geom. Topol."},{"issue":"3","key":"242_CR23","doi-asserted-by":"publisher","first-page":"610","DOI":"10.1007\/s00454-012-9399-3","volume":"47","author":"J Ma","year":"2012","unstructured":"Ma, J., Schlenker, J.-M.: Non-rigidity of spherical inversive distance circle packings. Discrete Comput. Geom. 47(3), 610\u2013617 (2012)","journal-title":"Discrete Comput. Geom."},{"key":"242_CR24","doi-asserted-by":"crossref","unstructured":"Marden, A., Rodin, B.: On Thurston\u2019s formulation and proof of Andreev\u2019s theorem. In: Computational Methods and Function Theory (Valpara\u00edso 1989). Lecture Notes in Math., vol. 1435, pp. 103\u2013115. Springer, Berlin (1990)","DOI":"10.1007\/BFb0087901"},{"issue":"1\u20133","key":"242_CR25","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1016\/0012-365X(93)90340-Y","volume":"117","author":"B Mohar","year":"1993","unstructured":"Mohar, B.: A polynomial time circle packing algorithm. Discrete Math. 117(1\u20133), 257\u2013263 (1993)","journal-title":"Discrete Math."},{"issue":"3","key":"242_CR26","doi-asserted-by":"publisher","first-page":"553","DOI":"10.2307\/2118572","volume":"139","author":"I Rivin","year":"1994","unstructured":"Rivin, I.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. 139(3), 553\u2013580 (1994)","journal-title":"Ann. Math."},{"issue":"2","key":"242_CR27","doi-asserted-by":"publisher","first-page":"349","DOI":"10.4310\/jdg\/1214441375","volume":"26","author":"B Rodin","year":"1987","unstructured":"Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping. J. Differ. Geom. 26(2), 349\u2013360 (1987)","journal-title":"J. Differ. Geom."},{"issue":"3","key":"242_CR28","doi-asserted-by":"publisher","first-page":"825","DOI":"10.5802\/aif.2279","volume":"57","author":"RKW Roeder","year":"2007","unstructured":"Roeder, R.K.W., Hubbard, J.H., Dunbar, W.D.: Andreev\u2019s theorem on hyperbolic polyhedra. Ann. Inst. Fourier (Grenoble) 57(3), 825\u2013882 (2007)","journal-title":"Ann. Inst. Fourier (Grenoble)"},{"key":"242_CR29","doi-asserted-by":"crossref","unstructured":"Schaeffer, D.G., Cain, J.W.: Ordinary Differential Equations: Basics and Beyond. Texts in Applied Mathematics, vol. 65. Springer, New York (2016)","DOI":"10.1007\/978-1-4939-6389-8"},{"issue":"3","key":"242_CR30","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/BF02773845","volume":"73","author":"O Schramm","year":"1991","unstructured":"Schramm, O.: Existence and uniqueness of packings with specified combinatorics. Israel J. Math. 73(3), 321\u2013341 (1991)","journal-title":"Israel J. Math."},{"issue":"3","key":"242_CR31","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1137\/0405034","volume":"5","author":"DD Sleator","year":"1992","unstructured":"Sleator, D.D., Tarjan, R.E., Thurston, W.P.: Short encodings of evolving structures. SIAM J. Discrete Math. 5(3), 428\u2013450 (1992)","journal-title":"SIAM J. Discrete Math."},{"key":"242_CR32","volume-title":"Introduction to Circle Packing","author":"K Stephenson","year":"2005","unstructured":"Stephenson, K.: Introduction to Circle Packing. Cambridge University Press, Cambridge (2005)"},{"key":"242_CR33","unstructured":"Thurston, W.P.: The Geometry and Topology of Three-Manifolds. Princeton University, Princeton (1980). (unpublished notes)"},{"key":"242_CR34","first-page":"26","volume":"46","author":"K Wagner","year":"1936","unstructured":"Wagner, K.: Bemerkungen zum Vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung 46, 26\u201332 (1936)","journal-title":"Jahresbericht der Deutschen Mathematiker-Vereinigung"},{"key":"242_CR35","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1016\/j.aim.2018.05.026","volume":"332","author":"X Xu","year":"2018","unstructured":"Xu, X.: Rigidity of inversive distance circle packings revisited. Adv. Math. 332, 476\u2013509 (2018)","journal-title":"Adv. Math."}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-020-00242-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-020-00242-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-020-00242-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,31]],"date-time":"2021-10-31T05:03:52Z","timestamp":1635656632000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-020-00242-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,14]]},"references-count":35,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["242"],"URL":"https:\/\/doi.org\/10.1007\/s00454-020-00242-8","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,14]]},"assertion":[{"value":"11 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 August 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 September 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}