17:
114:). However, the notation does not specify the configuration itself, only its type (the numbers of points, lines, and incidences). It also does not specify whether the configuration is purely combinatorial (an abstract incidence pattern of lines and points) or whether the points and lines of the configuration are realizable in the Euclidean plane or another standard geometry. The type (21
135:
smaller heptagons, one for each of the two lengths of diagonal; the sides of these smaller heptagons are the diagonals of the outer heptagon. Each of the two smaller heptagons has 14 diagonals, seven of which are shared with the other smaller heptagon. The seven shared diagonals are the remaining seven lines of the configuration.
134:
and its 14 interior diagonals. To complete the 21 points and lines of the configuration, these must be augmented by 14 more points and seven more lines. The remaining 14 points of the configuration are the points where pairs of equal-length diagonals of the heptagon cross each other. These form two
76:. Its original description by Klein in 1879 marked the first appearance in the mathematical literature of a 4-configuration, a system of points and lines with four points per line and four lines per point. In Klein's description, these points and lines belong to the
273:, and 21 interior points that do not belong to any tangent line. The 21 nonsecant lines and 21 interior points form an instance of the Grünbaum–Rigby configuration, meaning that again these points and lines have the same pattern of intersections.
290:
of the configuration includes symmetries that take any incident pair of points and lines to any other incident pair. The Grünbaum–Rigby configuration is an example of a polycyclic configuration, that is, a configuration with
285:
of this configuration, a system of points and lines with a point for every line of the configuration and a line for every point, and with the same point-line incidences, is the same configuration. The
926:
192:
107:). Their paper on it became the first of a series of works on configurations by Grünbaum, and contained the first published graphical depiction of a 4-configuration.
734:
522:
489:
271:
251:
219:
633:
603:
941:
118:) is highly ambiguous: there is an unknown but large number of (combinatorial) configurations of this type, 200 of which were listed by
885:
855:
983:
727:
701:, Mathematical Sciences Research Institute Publications, vol. 35, Cambridge, UK: Cambridge University Press, pp. 287–331,
444:
110:
In the notation of configurations, configurations with 21 points, 21 lines, 4 points per line and 4 lines per point are denoted (21
931:
962:
583:
517:
73:
32:
consisting of 21 points and 21 lines, with four points on each line and four lines through each point. Originally studied by
16:
905:
720:
587:
921:
936:
138:
The original construction of the Grünbaum–Rigby configuration by Klein viewed its points and lines as belonging to the
142:, rather than the Euclidean plane. In this space, the points and lines form the perspective centers and axes of the
624:
296:
143:
100:
53:
837:
139:
77:
37:
29:
870:
154:
880:
957:
825:
670:
150:. They have the same pattern of point-line intersections as the Euclidean version of the configuration.
890:
860:
800:
778:
900:
865:
845:
743:
875:
795:
687:
222:
195:
620:
575:
96:
49:
599:
461:
439:
159:
805:
757:
679:
642:
591:
531:
498:
480:
453:
69:
706:
654:
613:
563:
543:
510:
485:"On the Hessian configuration and its connection with the group of 360 plane collineations"
473:
702:
650:
609:
559:
539:
506:
469:
292:
282:
88:
45:
788:
287:
256:
236:
204:
130:
The Grünbaum–Rigby configuration can be constructed from the seven points of a regular
81:
457:
977:
691:
550:
Di Paola, Jane W.; Gropp, Harald (1989), "Hyperbolic graphs from hyperbolic planes",
147:
41:
783:
230:
697:
Klein, Felix (1999), "On the order-seven transformation of elliptic functions",
661:
226:
65:
33:
646:
535:
194:
has 57 points and 57 lines, and can be given coordinates based on the integers
850:
815:
762:
502:
465:
253:. Dually, there are 28 points where pairs of tangent lines meet, 8 points on
92:
87:
The geometric realisation of this configuration as points and lines in the
131:
666:"Ueber die Transformation siebenter Ordnung der elliptischen Functionen"
712:
683:
595:
233:
through a single point, and 21 nonsecant lines that are disjoint from
484:
199:
15:
84:
rather than the real-number coordinates of the
Euclidean plane.
716:
665:
259:
239:
207:
162:
299:
of points or lines has the same number of elements.
950:
914:
836:
771:
750:
265:
245:
213:
186:
523:Proceedings of the London Mathematical Society
490:Proceedings of the London Mathematical Society
64:The Grünbaum–Rigby configuration was known to
728:
422:
374:
119:
104:
8:
695:. Translated into English by Silvio Levy as
735:
721:
713:
634:Journal of the London Mathematical Society
258:
238:
206:
161:
568:
410:
362:
350:
338:
322:
314:
221:(the set of solutions to a two-variable
398:
326:
307:
95:, was only established much later, by
442:(2003), "Polycyclic configurations",
386:
318:
7:
91:, based on overlaying three regular
627:(1990), "The real configuration (21
586:, vol. 103, Providence, R.I.:
580:Configurations of points and lines
14:
445:European Journal of Combinatorics
80:, a space whose coordinates are
20:The Grünbaum-Rigby configuration
584:Graduate Studies in Mathematics
229:through pairs of its points, 8
44:, it was first realized in the
886:Cremona–Richmond configuration
181:
169:
1:
588:American Mathematical Society
458:10.1016/S0195-6698(03)00031-3
963:Kirkman's schoolgirl problem
896:Grünbaum–Rigby configuration
26:Grünbaum–Rigby configuration
856:Möbius–Kantor configuration
423:Boben & Pisanski (2003)
375:Grünbaum & Rigby (1990)
144:perspective transformations
120:Di Paola & Gropp (1989)
1000:
942:Bruck–Ryser–Chowla theorem
389:. See transl. p. 297.
984:Configurations (geometry)
932:Szemerédi–Trotter theorem
922:Sylvester–Gallai theorem
647:10.1112/jlms/s2-41.2.336
536:10.1112/plms/s3-46.1.117
198:7. In this space, every
140:complex projective plane
78:complex projective plane
38:complex projective plane
927:De Bruijn–Erdős theorem
871:Desargues configuration
187:{\displaystyle PG(2,7)}
155:finite projective plane
40:in connection with the
552:Congressus Numerantium
503:10.1112/plms/s2-4.1.54
267:
247:
215:
188:
21:
958:Design of experiments
671:Mathematische Annalen
268:
248:
216:
189:
19:
891:Kummer configuration
861:Pappus configuration
744:Incidence structures
520:(1983), "My graph",
257:
237:
205:
160:
60:History and notation
901:Klein configuration
881:Schläfli double six
866:Hesse configuration
846:Complete quadrangle
97:Branko Grünbaum
876:Reye configuration
684:10.1007/BF01677143
263:
243:
223:quadratic equation
211:
184:
22:
971:
970:
699:The Eightfold Way
637:, Second Series,
605:978-0-8218-4308-6
518:Coxeter, H. S. M.
493:, Second Series,
295:, such that each
266:{\displaystyle C}
246:{\displaystyle C}
225:modulo 7) has 28
214:{\displaystyle C}
24:In geometry, the
991:
806:Projective plane
758:Incidence matrix
737:
730:
723:
714:
709:
694:
657:
621:Grünbaum, Branko
616:
576:Grünbaum, Branko
566:
546:
526:, Third Series,
513:
476:
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408:
402:
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366:
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336:
330:
317:, p. 156);
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252:
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244:
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193:
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74:H. S. M. Coxeter
70:William Burnside
999:
998:
994:
993:
992:
990:
989:
988:
974:
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967:
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910:
832:
767:
763:Incidence graph
746:
741:
696:
660:
630:
619:
606:
596:10.1090/gsm/103
574:
569:Grünbaum (2009)
549:
516:
479:
440:Pisanski, Tomaž
437:
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429:
421:
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411:Grünbaum (2009)
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363:Grünbaum (2009)
361:
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351:Grünbaum (2009)
349:
345:
339:Grünbaum (2009)
337:
333:
323:Burnside (1907)
313:
309:
305:
293:cyclic symmetry
283:projective dual
279:
255:
254:
235:
234:
203:
202:
158:
157:
128:
117:
113:
89:Euclidean plane
82:complex numbers
62:
50:Branko Grünbaum
46:Euclidean plane
28:is a symmetric
12:
11:
5:
997:
995:
987:
986:
976:
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969:
968:
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952:
948:
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937:Beck's theorem
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838:Configurations
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789:Steiner system
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751:Representation
748:
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717:
711:
710:
678:(3): 428–471,
658:
641:(2): 336–346,
628:
617:
604:
572:
567:. As cited by
547:
530:(1): 117–136,
514:
477:
452:(4): 431–457,
438:Boben, Marko;
433:
430:
428:
427:
415:
413:, p. 363.
403:
399:Coxeter (1983)
391:
379:
367:
355:
343:
341:, p. 156.
331:
327:Coxeter (1983)
315:Grünbaum (2009
306:
304:
301:
288:symmetry group
278:
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262:
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127:
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61:
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13:
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9:
6:
4:
3:
2:
996:
985:
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938:
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847:
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841:
839:
835:
827:
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817:
814:
813:
812:Graph theory
811:
807:
804:
802:
799:
798:
797:
794:
790:
787:
785:
782:
781:
780:
779:Combinatorics
777:
776:
774:
770:
764:
761:
759:
756:
755:
753:
749:
745:
738:
733:
731:
726:
724:
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715:
708:
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693:
689:
685:
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677:
673:
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667:
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508:
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424:
419:
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412:
407:
404:
400:
395:
392:
388:
383:
380:
376:
371:
368:
365:, p. 53.
364:
359:
356:
353:, p. 13.
352:
347:
344:
340:
335:
332:
328:
324:
320:
316:
311:
308:
302:
300:
298:
294:
289:
284:
276:
274:
260:
240:
232:
231:tangent lines
228:
224:
208:
201:
197:
178:
175:
172:
166:
163:
156:
151:
149:
148:Klein quartic
145:
141:
136:
133:
125:
123:
121:
108:
106:
102:
99: and
98:
94:
90:
85:
83:
79:
75:
71:
67:
59:
57:
55:
54:John F. Rigby
51:
47:
43:
42:Klein quartic
39:
35:
31:
30:configuration
27:
18:
951:Applications
895:
784:Block design
698:
675:
669:
662:Klein, Felix
638:
632:
625:Rigby, J. F.
579:
555:
551:
527:
521:
494:
488:
481:Burnside, W.
449:
443:
418:
406:
394:
387:Klein (1879)
382:
370:
358:
346:
334:
319:Klein (1879)
310:
280:
227:secant lines
152:
137:
129:
126:Construction
109:
86:
63:
25:
23:
822:Statistics
101:J. F. Rigby
66:Felix Klein
34:Felix Klein
851:Fano plane
816:Hypergraph
432:References
277:Properties
93:heptagrams
801:Incidence
692:121407539
558:: 23–43,
497:: 54–71,
466:0195-6698
978:Category
915:Theorems
826:Blocking
796:Geometry
664:(1879),
578:(2009),
483:(1907),
132:heptagon
707:1722419
655:1067273
614:2510707
564:0995852
544:0684825
511:1576105
474:1975946
146:of the
103: (
36:in the
772:Fields
705:
690:
653:
612:
602:
562:
542:
509:
472:
464:
196:modulo
72:, and
688:S2CID
303:Notes
297:orbit
200:conic
906:Dual
631:)",
600:ISBN
462:ISSN
281:The
153:The
105:1990
52:and
680:doi
643:doi
592:doi
532:doi
499:doi
454:doi
48:by
980::
703:MR
686:,
676:14
674:,
668:,
651:MR
649:,
639:41
623:;
610:MR
608:,
598:,
590:,
582:,
560:MR
556:68
554:,
540:MR
538:,
528:46
507:MR
505:,
487:,
470:MR
468:,
460:,
450:24
448:,
325:;
321:;
122:.
68:,
56:.
736:e
729:t
722:v
682::
645::
629:4
594::
571:.
534::
501::
495:4
456::
425:.
401:.
377:.
329:.
261:C
241:C
209:C
182:)
179:7
176:,
173:2
170:(
167:G
164:P
116:4
112:4
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