17:
315:
375:. Each successive circle after these first two is tangent to the central unit circle and to the two most recently added circles; see the illustration for the first six circles (including the two halfplanes) constructed in this way. The first
366:
circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lemma. It begins by sandwiching the unit circle between two parallel halfplanes; in
227:
140:
88:
interior-disjoint circles, all tangent to it, with consecutive circles in the ring tangent to each other. Then the minimum radius of any circle in the ring is at least the
167:
459:
Beyond its original application to conformal mapping, the circle packing theorem and the ring lemma play key roles in a proof by
Keszegh, Pach, and Pálvölgyi that
220:
417:
393:
356:
187:
86:
66:
324:
733:
760:
498:
94:
734:
Graph
Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers
310:{\textstyle \displaystyle 1,{\frac {1}{4}},{\frac {1}{12}},{\frac {1}{33}},{\frac {1}{88}},{\frac {1}{232}},\dots }
804:
488:
399:
to be the same as the radius specified in the ring lemma. This construction can be perturbed to a ring of
419:
finite circles, without additional tangencies, whose minimum radius is arbitrarily close to this bound.
799:
68:
be any integer greater than or equal to three. Suppose that the unit circle is surrounded by a ring of
363:
789:
358:
that exactly meet the bound of the ring lemma, showing that it is tight. The construction allows
448:
25:
517:
737:, Lecture Notes in Computer Science, vol. 6502, Heidelberg: Springer, pp. 293–304,
513:
396:
359:
770:
727:; Pálvölgyi, Dömötör (2011), "Drawing planar graphs of bounded degree with few slopes", in
710:
676:
626:
588:
554:
508:
145:
8:
444:
199:
738:
592:
402:
378:
368:
341:
172:
71:
51:
756:
596:
494:
372:
395:
circles of this construction form a ring, whose minimum radius can be calculated by
748:
698:
662:
576:
542:
436:
190:
650:
794:
766:
752:
706:
672:
646:
622:
584:
550:
504:
432:
29:
702:
580:
546:
338:
An infinite sequence of circles can be constructed, containing rings for each
783:
728:
667:
440:
89:
724:
642:
464:
460:
428:
490:
Introduction to Circle
Packing: The Theory of Discrete Analytic Functions
37:
427:
A version of the ring lemma with a weaker bound was first proven by
743:
689:
Hansen, Lowell J. (1988), "On the Rodin and
Sullivan ring lemma",
447:
for the tightest possible lower bound, and Dov
Aharonov found a
567:
Vasilis, Jonatan (2011), "The ring lemma in three dimensions",
610:
533:
Aharonov, Dov (1997), "The sharp constant in the ring lemma",
439:'s conjecture that circle packings can be used to approximate
16:
651:"The convergence of circle packings to the Riemann mapping"
330:
Generalizations to three-dimensional space are also known.
319:
371:, these are considered to be tangent to each other at the
611:"Geometric sequences of discs in the Apollonian packing"
20:
Construction showing the tight bound for the ring lemma
230:
40:
on the sizes of adjacent circles in a circle packing.
405:
381:
344:
231:
202:
175:
148:
97:
74:
54:
722:
608:
411:
387:
350:
309:
214:
181:
161:
134:
80:
60:
781:
641:
463:of bounded degree can be drawn with bounded
486:
742:
666:
512:; see especially Lemma 8.2 (Ring Lemma),
532:
15:
566:
135:{\displaystyle {\frac {1}{F_{2n-3}-1}}}
782:
688:
609:Aharonov, D.; Stephenson, K. (1997),
637:
635:
196:The sequence of minimum radii, from
528:
526:
482:
480:
13:
716:
516:, and Appendix B, The Ring Lemma,
14:
816:
632:
602:
682:
655:Journal of Differential Geometry
560:
523:
477:
454:
333:
493:, Cambridge University Press,
1:
470:
753:10.1007/978-3-642-18469-7_27
731:; Cornelsen, Sabine (eds.),
487:Stephenson, Kenneth (2005),
43:
7:
10:
821:
435:as part of their proof of
422:
703:10.1080/17476938808814284
581:10.1007/s10711-010-9545-0
547:10.1080/17476939708815009
443:. Lowell Hansen gave a
369:the geometry of circles
805:Geometric inequalities
668:10.4310/jdg/1214441375
449:closed-form expression
413:
389:
352:
311:
216:
183:
163:
136:
82:
62:
48:The lemma states: Let
21:
414:
390:
353:
312:
217:
184:
164:
162:{\displaystyle F_{i}}
137:
83:
63:
19:
451:for the same bound.
403:
379:
362:to be considered as
342:
228:
200:
173:
146:
95:
72:
52:
569:Geometriae Dedicata
445:recurrence relation
215:{\displaystyle n=3}
24:In the geometry of
409:
397:Descartes' theorem
385:
348:
307:
306:
212:
179:
159:
132:
78:
58:
22:
800:Fibonacci numbers
762:978-3-642-18468-0
723:Keszegh, Balázs;
691:Complex Variables
535:Complex Variables
500:978-0-521-82356-2
412:{\displaystyle n}
388:{\displaystyle n}
373:point at infinity
351:{\displaystyle n}
298:
285:
272:
259:
246:
182:{\displaystyle i}
130:
81:{\displaystyle n}
61:{\displaystyle n}
812:
774:
773:
746:
720:
714:
713:
686:
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679:
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647:Sullivan, Dennis
639:
630:
629:
615:Algebra i Analiz
606:
600:
599:
564:
558:
557:
530:
521:
511:
484:
437:William Thurston
418:
416:
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410:
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299:
291:
286:
278:
273:
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252:
247:
239:
221:
219:
218:
213:
191:Fibonacci number
188:
186:
185:
180:
168:
166:
165:
160:
158:
157:
141:
139:
138:
133:
131:
129:
122:
121:
99:
87:
85:
84:
79:
67:
65:
64:
59:
820:
819:
815:
814:
813:
811:
810:
809:
780:
779:
778:
777:
763:
721:
717:
687:
683:
640:
633:
607:
603:
565:
561:
531:
524:
501:
485:
478:
473:
457:
433:Dennis Sullivan
425:
404:
401:
400:
380:
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376:
343:
340:
339:
336:
328:
318:
290:
277:
264:
251:
238:
229:
226:
225:
201:
198:
197:
174:
171:
170:
153:
149:
147:
144:
143:
108:
104:
103:
98:
96:
93:
92:
73:
70:
69:
53:
50:
49:
46:
30:Euclidean plane
26:circle packings
12:
11:
5:
818:
808:
807:
802:
797:
792:
790:Circle packing
776:
775:
761:
729:Brandes, Ulrik
715:
681:
661:(2): 349–360,
631:
621:(3): 104–140,
601:
559:
541:(1–4): 27–31,
522:
499:
475:
474:
472:
469:
456:
453:
441:conformal maps
424:
421:
408:
384:
347:
335:
332:
305:
302:
297:
294:
289:
284:
281:
276:
271:
268:
263:
258:
255:
250:
245:
242:
237:
234:
224:
211:
208:
205:
178:
156:
152:
128:
125:
120:
117:
114:
111:
107:
102:
77:
57:
45:
42:
9:
6:
4:
3:
2:
817:
806:
803:
801:
798:
796:
793:
791:
788:
787:
785:
772:
768:
764:
758:
754:
750:
745:
740:
736:
735:
730:
726:
719:
712:
708:
704:
700:
696:
692:
685:
678:
674:
669:
664:
660:
656:
652:
648:
644:
638:
636:
628:
624:
620:
616:
612:
605:
598:
594:
590:
586:
582:
578:
574:
570:
563:
556:
552:
548:
544:
540:
536:
529:
527:
519:
515:
510:
506:
502:
496:
492:
491:
483:
481:
476:
468:
466:
462:
461:planar graphs
452:
450:
446:
442:
438:
434:
430:
420:
406:
398:
382:
374:
370:
365:
361:
345:
331:
326:
321:
303:
300:
295:
292:
287:
282:
279:
274:
269:
266:
261:
256:
253:
248:
243:
240:
235:
232:
223:
209:
206:
203:
194:
192:
176:
154:
150:
126:
123:
118:
115:
112:
109:
105:
100:
91:
90:unit fraction
75:
55:
41:
39:
35:
31:
27:
18:
732:
718:
697:(1): 23–30,
694:
690:
684:
658:
654:
618:
614:
604:
572:
568:
562:
538:
534:
489:
465:slope number
458:
455:Applications
429:Burton Rodin
426:
337:
334:Construction
329:
195:
47:
33:
23:
725:Pach, János
643:Rodin, Burt
518:pp. 318–321
38:lower bound
784:Categories
471:References
364:degenerate
360:halfplanes
317:(sequence
34:ring lemma
744:1009.1315
597:120113578
575:: 51–62,
514:pp. 73–74
304:…
222:, begins
124:−
116:−
44:Statement
649:(1987),
36:gives a
771:2781274
711:0946096
677:0906396
627:1466797
589:2795235
555:1624890
509:2131318
423:History
323:in the
320:A027941
169:is the
28:in the
795:Lemmas
769:
759:
709:
675:
625:
595:
587:
553:
507:
497:
142:where
32:, the
739:arXiv
593:S2CID
757:ISBN
495:ISBN
431:and
325:OEIS
749:doi
699:doi
663:doi
577:doi
573:152
543:doi
296:232
189:th
786::
767:MR
765:,
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747:,
707:MR
705:,
695:10
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673:MR
671:,
659:26
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645:;
634:^
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665::
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520:.
407:n
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301:,
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233:1
210:3
207:=
204:n
177:i
155:i
151:F
127:1
119:3
113:n
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101:1
76:n
56:n
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