17:
158:
contains each of the points derived from its three duads. Thus, the duads and synthemes of the abstract configuration correspond one-for-one, in an incidence-preserving way, with these 15 points and 15 lines derived from the original six points, which form a realization of the configuration. The same realization may be projected into
Euclidean space or the Euclidean plane.
141:: it is possible to exchange points for lines while preserving all the incidences of the configuration. This duality gives the Tutte–Coxeter graph additional symmetries beyond those of the Cremona–Richmond configuration, which swap the two sides of its bipartition. These symmetries correspond to the
157:
through the other four points; thus, the duads of the six points correspond one-for-one with these 15 derived points. Any three duads that together form a syntheme determine a line, the intersection line of the three hyperplanes containing two of the three duads in the syntheme, and this line
281:, but Sylvester treats these systems of pairs and partitions in the context of a more general study of tuples and partitions of sets, does not reserve special attention to the case of a six-element set, and does not associate any geometric meaning to the sets.
123:. Identified in this way, a point of the configuration is incident to a line of the configuration if and only if the duad corresponding to the point is one of the three pairs in the syntheme corresponding to the line.
188:
in the set of all 27 lines on a cubic) and 15 tangent planes, with three lines in each plane and three planes through each line. Intersecting these lines and planes by another plane results in a 15
610:"An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface"
113:
989:
130:
of all permutations of the six elements underlying this system of duads and synthemes acts as a symmetry group of the
Cremona–Richmond configuration, and gives the
119:. Similarly, the lines of the configuration may be identified with the 15 ways of partitioning the same six elements into three pairs; these partitions are called
797:
631:"On the distribution of surfaces of the third order into species, in reference to the absence or presence of singular points, and the reality of their lines"
134:
group of the configuration. Every flag of the configuration (an incident point-line pair) can be taken to every other flag by a symmetry in this group.
547:
1004:
958:
393:
918:
153:
Any six points in general position in four-dimensional space determine 15 points where a line through two of the points intersects the
32:
of 15 lines and 15 points, having 3 points on each line and 3 lines through each point, and containing no triangles. It was studied by
1046:
790:
463:
994:
161:
The
Cremona–Richmond configuration also has a one-parameter family of realizations in the plane with order-five cyclic symmetry.
1025:
535:
458:
424:
224:
used the four-dimensional realization of this configuration as the frontispiece for two volumes of his 1922–1925 textbook,
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783:
539:
138:
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999:
711:
247:; perhaps due to some mistakes in his work, the contemporaneous contribution of Martinetti fell into obscurity.
900:
585:
209:
41:
29:
57:
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943:
185:
49:
232:
also rediscovered the same configuration, and found a realization of it with order-five cyclic symmetry.
71:
1020:
888:
953:
923:
863:
841:
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765:
196:
configuration. The specific incidence pattern of Schläfli's lines and planes was later published by
963:
928:
908:
806:
752:
16:
938:
858:
480:
142:
705:
Zacharias, Max (1951), "Streifzüge im Reich der
Konfigurationen: Eine Reyesche Konfiguration (15
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553:
446:
412:
365:
127:
851:
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204:). The observation that the resulting configuration contains no triangles was made by
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513:
492:
197:
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519:
Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal
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221:
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679:
407:
154:
53:
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385:
340:
757:
235:
The name of the configuration comes from the studies of it by
Cremona (
68:
The points of the
Cremona–Richmond configuration may be identified with the
647:
476:
495:(1868), "Mémoire de géométrie pure sur les surfaces du troisieme ordre",
115:
unordered pairs of elements of a six-element set; these pairs are called
775:
577:
220:
found a description of the configuration as a self-inscribed polygon.
484:
661:"Elementary researches in the analysis of combinatorial aggregation"
461:(1958), "Twelve points in PG(5,3) with 95040 self-transformations",
15:
779:
313:
This history and most of the references in it are drawn from
687:
Visconti, E. (1916), "Sulle configurazioni piane atrigone",
564:
Martinetti, V. (1886), "Sopra alcune configurazioni piane",
698:
506:
314:
298:
590:"On the figure of six points in space of four dimensions."
208:, and the same configuration also appears in the work of
386:"Small triangle-free configurations of points and lines"
427:(1950), "Self-dual configurations and regular graphs",
82:
184:
containing sets of 15 real lines (complementary to a
74:
1013:
977:
899:
834:
813:
107:
522:, Atti della R. Accademia dei Lincei, vol. 1
277:. The terminology of duads and synthemes is from
635:Philosophical Transactions of the Royal Society
791:
429:Bulletin of the American Mathematical Society
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145:of the symmetric group on six elements.
45:
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201:
37:
771:Image ofCremona–Richmond configuration
766:Image ofCremona–Richmond configuration
709:), Stern- und Kettenkonfigurationen",
566:Annali di Matematica Pura ed Applicata
309:
307:
137:The Cremona–Richmond configuration is
689:Giornale di Matematiche di Battaglini
7:
266:
264:
262:
260:
538:, vol. 103, Providence, R.I.:
394:Discrete and Computational Geometry
108:{\displaystyle 15={\tbinom {6}{2}}}
532:Configurations of points and lines
464:Proceedings of the Royal Society A
89:
20:The Cremona–Richmond configuration
14:
348:European Journal of Combinatorics
317:. The reference to Baker is from
753:"Cremona–Richmond Configuration"
536:Graduate Studies in Mathematics
442:10.1090/S0002-9904-1950-09407-5
949:Cremona–Richmond configuration
26:Cremona–Richmond configuration
1:
540:American Mathematical Society
361:10.1016/S0195-6698(03)00031-3
1026:Kirkman's schoolgirl problem
959:Grünbaum–Rigby configuration
210:Herbert William Richmond
919:Möbius–Kantor configuration
341:"Polycyclic configurations"
295:Boben & Pisanski (2003)
52:with parameters (2,2). Its
1063:
1005:Bruck–Ryser–Chowla theorem
614:Quart. J. Pure Appl. Math.
1047:Configurations (geometry)
995:Szemerédi–Trotter theorem
712:Mathematische Nachrichten
680:10.1080/14786444408644856
408:10.1007/s00454-005-1224-9
384:; Žitnik, Arjana (2006),
985:Sylvester–Gallai theorem
725:10.1002/mana.19510050602
990:De Bruijn–Erdős theorem
934:Desargues configuration
648:10.1098/rstl.1863.0010
477:10.1098/rspa.1958.0184
226:Principles of Geometry
109:
50:generalized quadrangle
21:
1021:Design of experiments
497:J. Reine Angew. Math.
110:
19:
954:Kummer configuration
924:Pappus configuration
807:Incidence structures
72:
24:In mathematics, the
964:Klein configuration
944:Schläfli double six
929:Hesse configuration
909:Complete quadrangle
699:Boben et al. (2006)
507:Boben et al. (2006)
315:Boben et al. (2006)
299:Boben et al. (2006)
186:Schläfli double six
170:Ludwig Schläfli
143:outer automorphisms
58:Tutte–Coxeter graph
939:Reye configuration
749:Weisstein, Eric W.
578:10.1007/BF02420733
105:
103:
22:
1034:
1033:
549:978-0-8218-4308-6
471:(1250): 279–293,
459:Coxeter, H. S. M.
425:Coxeter, H. S. M.
206:Martinetti (1886)
198:Luigi Cremona
96:
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869:Projective plane
821:Incidence matrix
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657:Sylvester, J. J.
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382:Pisanski, Tomaž
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245:Richmond (1900)
218:Visconti (1916)
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128:symmetric group
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901:Configurations
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814:Representation
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741:External links
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572:(1): 161–192,
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505:. As cited by
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401:(3): 405–427,
376:Boben, Marko;
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354:(4): 431–457,
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319:Coxeter (1950)
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275:Coxeter (1958)
271:Coxeter (1950)
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30:configuration
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1014:Applications
948:
847:Block design
756:
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692:
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671:
670:, Series 3,
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638:
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627:Schläfli, L.
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606:Schläfli, L.
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568:, Series 2,
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337:Pisanski, T.
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132:automorphism
125:
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885:Statistics
719:: 329–345,
674:: 285–295,
641:: 193–241,
514:Cremona, L.
493:Cremona, L.
435:: 413–455,
335:Boben, M.;
222:H. F. Baker
149:Realization
48:). It is a
914:Fano plane
879:Hypergraph
668:Phil. Mag.
329:References
155:hyperplane
54:Levi graph
864:Incidence
758:MathWorld
600:: 125–160
594:Quart. J.
139:self-dual
121:synthemes
1041:Category
978:Theorems
889:Blocking
859:Geometry
659:(1844),
629:(1863),
608:(1858),
588:(1900),
530:(2009),
516:(1877),
339:(2003),
180:) found
64:Symmetry
42:Richmond
733:0043473
695:: 27–41
558:2510707
503:: 1–133
451:0038078
417:2202110
370:1975946
212: (
200: (
172: (
165:History
56:is the
44: (
36: (
34:Cremona
835:Fields
731:
556:
546:
485:100667
483:
449:
415:
368:
243:) and
40:) and
664:(PDF)
481:JSTOR
389:(PDF)
344:(PDF)
251:Notes
117:duads
28:is a
969:Dual
544:ISBN
241:1877
237:1868
214:1900
202:1868
178:1863
174:1858
126:The
46:1900
38:1877
721:doi
676:doi
643:doi
639:153
574:doi
473:doi
469:247
437:doi
403:doi
356:doi
216:).
1043::
755:,
751:,
729:MR
727:,
715:,
693:54
691:,
672:24
666:,
637:,
633:,
616:,
612:,
598:31
596:,
592:,
570:14
554:MR
552:,
542:,
534:,
501:68
499:,
479:,
467:,
447:MR
445:,
433:56
431:,
413:MR
411:,
399:35
397:,
391:,
380:;
366:MR
364:,
352:24
350:,
346:,
306:^
297:;
293:;
273:;
259:^
239:,
228:.
192:15
176:,
76:15
60:.
799:e
792:t
785:v
736:.
723::
717:5
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79:=
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