28:
20:
263:. The four ovals can be grouped into six different pairs of ovals; for each pair of ovals there are four bitangents touching both ovals in the pair, two that separate the two ovals, and two that do not. Additionally, each oval bounds a nonconvex region of the plane and has one bitangent spanning the nonconvex portion of its boundary.
448:
as the coordinates of a line in the same finite projective plane; the condition that the sum is odd is equivalent to requiring that the point and the line do not touch each other, and there are 28 different pairs of a point and a line that do not touch.
452:
The points and lines of the Fano plane that are disjoint from a non-incident point-line pair form a triangle, and the bitangents of a quartic have been considered as being in correspondence with the 28 triangles of the Fano plane. The
243:
342:
423:
70:) As Plücker showed, the number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real bitangents can be formed by the
926:
259:
Like the examples of Plücker and of Blum and
Guinand, the Trott curve has four separated ovals, the maximum number for a curve of degree four, and hence is an
461:, in which the triangles of the Fano plane are represented by 6-cycles. The 28 6-cycles of the Heawood graph in turn correspond to the 28 vertices of the
124:
284:
639:
881:
805:
990:
747:
704:
357:
23:
The Trott curve and seven of its bitangents. The others are symmetric with respect to 90° rotations through the origin.
479:
The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a canonic
260:
48:
985:
437:
756:
111:
275:
to a quartic curve has 28 real ordinary double points, dual to the 28 bitangents of the primal curve.
896:
Theorie der algebraischen Curven: gegrundet auf eine neue
Behandlungsweise der analytischen Geometrie
492:
473:
761:
645:
71:
40:
852:
832:
791:
782:
620:
602:
505:
484:
82:
gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a
920:
891:
877:
801:
469:
63:
278:
The 28 bitangents of a quartic may also be placed in correspondence with symbols of the form
904:
869:
842:
766:
713:
612:
87:
955:
815:
778:
674:
86:; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the
951:
811:
774:
488:
480:
249:
56:
660:
670:
27:
468:
The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2
19:
979:
738:, Andrew Russell Forsyth, ed., The University Press, 1896, vol. 11, pp. 221–223.
725:
462:
458:
83:
856:
786:
847:
238:{\displaystyle \displaystyle 144(x^{4}+y^{4})-225(x^{2}+y^{2})+350x^{2}y^{2}+81=0.}
51:, but it is possible to define quartic curves for which all 28 of these lines have
47:
lines, lines that are tangent to the curve in two places. These lines exist in the
666:
624:
873:
800:, MSRI Publications, vol. 35, Cambridge University Press, pp. 115–131,
936:
742:
52:
36:
962:
Trott, Michael (1997), "Applying
GroebnerBasis to Three Problems in Geometry",
864:
McKay, John; Sebbar, Abdellah (2007). "Replicable
Functions: An Introduction".
454:
441:
272:
115:
337:{\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\\end{bmatrix}}}
253:
44:
717:
823:
Manivel, L. (2006), "Configurations of lines and models of Lie algebras",
432:, but only 28 of these choices produce an odd sum. One may also interpret
62:
An explicit quartic with twenty-eight real bitangents was first given by
770:
661:
Lecture 2: Symplectization, Complexification and
Mathematical Trinities
75:
702:
Blum, R.; Guinand, A. P. (1964). "A quartic with 28 real bitangents".
616:
593:
Dejter, Italo J. (2011), "From the
Coxeter graph to the Klein graph",
837:
607:
26:
18:
16:
28 lines which touch a general quartic plane curve in two places
658:
Arnold 1997, p. 13 – Arnold, Vladimir, 1997, Toronto
Lectures,
102:, another curve with 28 real bitangents, is the set of points (
907:(1876), "Zur Theorie der Abel'schen Funktionen für den Fall
78:
with fixed axis lengths, tangent to two non-parallel lines.
495:, and can be related to many further objects, including E
248:
These points form a nonsingular quartic curve that has
418:{\displaystyle ad+be+cf=1\ (\operatorname {mod} \ 2).}
293:
360:
287:
128:
127:
866:
Frontiers in Number Theory, Physics, and
Geometry II
944:Commentarii Mathematici Universitatis Sancti Pauli
736:The collected mathematical papers of Arthur Cayley
417:
336:
237:
55:as their coordinates and therefore belong to the
937:"Weierstrass transformations and cubic surfaces"
745:(1982), "From the history of a simple group",
8:
686:
537:
925:: CS1 maint: location missing publisher (
728:(1879), "On the bitangents of a quartic",
846:
836:
760:
606:
359:
288:
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216:
206:
187:
174:
152:
139:
126:
665:June 1997 (last updated August, 1998).
580:
561:
517:
67:
31:The Trott curve with all 28 bitangents.
918:
565:
79:
964:Mathematica in Education and Research
576:
574:
549:
252:three and that has twenty-eight real
7:
525:
14:
638:le Bruyn, Lieven (17 June 2008),
267:Connections to other structures
848:10.1016/j.jalgebra.2006.04.029
748:The Mathematical Intelligencer
705:Canadian Mathematical Bulletin
409:
394:
351:are all zero or one and where
193:
167:
158:
132:
1:
874:10.1007/978-3-540-30308-4_10
730:Salmon's Higher Plane Curves
915:, Leipzig, pp. 456–472
35:In the theory of algebraic
1007:
796:Levy, Silvio, ed. (1999),
538:Blum & Guinand (1964)
491:, specifically a form of
457:of the Fano plane is the
428:There are 64 choices for
90:in the projective plane.
935:Shioda, Tetsuji (1995),
49:complex projective plane
991:Real algebraic geometry
898:, Berlin: Adolph Marcus
687:McKay & Sebbar 2007
595:Journal of Graph Theory
438:homogeneous coordinates
718:10.4153/cmb-1964-038-6
419:
338:
239:
32:
24:
931:. As cited by Cayley.
474:theta characteristics
420:
339:
240:
30:
22:
868:. pp. 373–386.
493:McKay correspondence
483:of genus 4, form a "
472:, and to the 28 odd
358:
285:
125:
41:quartic plane curve
825:Journal of Algebra
771:10.1007/BF03023483
732:, pp. 387–389
641:Arnold's trinities
503:, as discussed at
487:" in the sense of
440:of a point of the
415:
334:
328:
235:
234:
33:
25:
911: = 3",
905:Riemann, G. F. B.
883:978-3-540-30307-7
798:The Eightfold Way
617:10.1002/jgt.20597
470:del Pezzo surface
405:
393:
110:) satisfying the
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430:a, b, c, d, e, f
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349:a, b, c, d, e, f
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88:line at infinity
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57:Euclidean plane
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12:
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5:
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986:Quartic curves
978:
977:
974:
973:
959:
950:(1): 109–128,
932:
901:
888:
882:
861:
831:(1): 457–486,
820:
806:
739:
726:Cayley, Arthur
722:
712:(3): 399–404.
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678:
651:
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581:Manivel (2006)
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562:Riemann (1876)
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74:of centers of
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807:0-521-66066-1
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707:
706:
700:
699:
695:
688:
682:
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676:
672:
668:
664:
662:
655:
652:
648:on 2011-04-11
647:
643:
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626:
622:
618:
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566:Cayley (1879)
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531:
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518:
512:
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486:
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477:
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471:
466:
464:
463:Coxeter graph
460:
459:Heawood graph
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84:cubic surface
81:
80:Shioda (1995)
77:
73:
69:
65:
60:
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54:
50:
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42:
38:
29:
21:
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908:
895:
865:
838:math/0507118
828:
824:
797:
755:(2): 59–67,
752:
746:
743:Gray, Jeremy
735:
729:
709:
703:
681:
659:
654:
646:the original
640:
633:
598:
594:
588:
557:
550:Trott (1997)
545:
533:
520:
504:
481:sextic curve
478:
467:
451:
427:
346:
277:
270:
258:
247:
107:
103:
99:
97:
61:
53:real numbers
39:, a general
37:plane curves
34:
892:Plücker, J.
526:Gray (1982)
100:Trott curve
980:Categories
970:(1): 15–28
913:Ges. Werke
696:References
671:PostScript
455:Levi graph
442:Fano plane
273:dual curve
254:bitangents
116:polynomial
792:Reprinted
757:CiteSeerX
608:1002.1960
524:See e.g.
506:trinities
401:
162:−
118:equation
45:bitangent
921:citation
894:(1839),
857:17374533
787:14602496
689:, p. 11)
76:ellipses
956:1336422
816:1722415
779:0672918
601:: 1–9,
485:trinity
446:d, e, f
436:as the
434:a, b, c
261:M-curve
94:Example
66: (
64:Plücker
43:has 28
954:
880:
855:
814:
804:
785:
777:
759:
625:754481
623:
404:
392:
347:where
112:degree
940:(PDF)
853:S2CID
833:arXiv
783:S2CID
734:. In
621:S2CID
603:arXiv
513:Notes
499:and E
250:genus
114:four
72:locus
927:link
878:ISBN
802:ISBN
444:and
271:The
98:The
68:1839
870:doi
843:doi
829:304
794:in
767:doi
714:doi
675:PDF
667:TeX
613:doi
398:mod
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