39:
31:
274:. 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.
459:
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.
463:
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
254:
353:
434:
81:) 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
937:
270:
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
472:, 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
135:
295:
650:
892:
816:
1001:
758:
715:
368:
34:
The Trott curve and seven of its bitangents. The others are symmetric with respect to 90° rotations through the origin.
490:
The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a canonic
271:
59:
996:
448:
767:
122:
286:
to a quartic curve has 28 real ordinary double points, dual to the 28 bitangents of the primal curve.
907:
Theorie der algebraischen Curven: gegrundet auf eine neue
Behandlungsweise der analytischen Geometrie
503:
484:
772:
656:
82:
51:
863:
843:
802:
793:
631:
613:
516:
495:
93:
gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a
931:
902:
888:
812:
480:
74:
289:
The 28 bitangents of a quartic may also be placed in correspondence with symbols of the form
915:
880:
853:
777:
724:
623:
98:
966:
826:
789:
685:
97:; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the
962:
822:
785:
499:
491:
260:
67:
671:
681:
38:
479:
The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2
30:
990:
749:, Andrew Russell Forsyth, ed., The University Press, 1896, vol. 11, pp. 221–223.
736:
473:
469:
94:
867:
797:
858:
249:{\displaystyle \displaystyle 144(x^{4}+y^{4})-225(x^{2}+y^{2})+350x^{2}y^{2}+81=0.}
62:, but it is possible to define quartic curves for which all 28 of these lines have
58:
lines, lines that are tangent to the curve in two places. These lines exist in the
677:
635:
884:
811:, MSRI Publications, vol. 35, Cambridge University Press, pp. 115–131,
947:
753:
63:
47:
973:
Trott, Michael (1997), "Applying
GroebnerBasis to Three Problems in Geometry",
875:
McKay, John; Sebbar, Abdellah (2007). "Replicable
Functions: An Introduction".
465:
452:
283:
126:
348:{\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\\end{bmatrix}}}
264:
55:
728:
17:
834:
Manivel, L. (2006), "Configurations of lines and models of Lie algebras",
443:, but only 28 of these choices produce an odd sum. One may also interpret
73:
An explicit quartic with twenty-eight real bitangents was first given by
781:
672:
Lecture 2: Symplectization, Complexification and
Mathematical Trinities
86:
713:
Blum, R.; Guinand, A. P. (1964). "A quartic with 28 real bitangents".
627:
604:
Dejter, Italo J. (2011), "From the
Coxeter graph to the Klein graph",
848:
618:
37:
29:
27:
28 lines which touch a general quartic plane curve in two places
669:
Arnold 1997, p. 13 – Arnold, Vladimir, 1997, Toronto
Lectures,
113:, another curve with 28 real bitangents, is the set of points (
918:(1876), "Zur Theorie der Abel'schen Funktionen für den Fall
89:
with fixed axis lengths, tangent to two non-parallel lines.
506:, and can be related to many further objects, including E
259:
These points form a nonsingular quartic curve that has
429:{\displaystyle ad+be+cf=1\ (\operatorname {mod} \ 2).}
304:
371:
298:
139:
138:
877:
Frontiers in Number Theory, Physics, and
Geometry II
955:Commentarii Mathematici Universitatis Sancti Pauli
747:The collected mathematical papers of Arthur Cayley
428:
347:
248:
66:as their coordinates and therefore belong to the
948:"Weierstrass transformations and cubic surfaces"
756:(1982), "From the history of a simple group",
8:
697:
548:
936:: CS1 maint: location missing publisher (
739:(1879), "On the bitangents of a quartic",
857:
847:
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227:
217:
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163:
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137:
676:June 1997 (last updated August, 1998).
591:
572:
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42:The Trott curve with all 28 bitangents.
929:
576:
90:
975:Mathematica in Education and Research
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585:
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263:three and that has twenty-eight real
7:
536:
25:
649:le Bruyn, Lieven (17 June 2008),
278:Connections to other structures
859:10.1016/j.jalgebra.2006.04.029
759:The Mathematical Intelligencer
716:Canadian Mathematical Bulletin
420:
405:
362:are all zero or one and where
204:
178:
169:
143:
1:
885:10.1007/978-3-540-30308-4_10
741:Salmon's Higher Plane Curves
926:, Leipzig, pp. 456–472
46:In the theory of algebraic
1018:
807:Levy, Silvio, ed. (1999),
549:Blum & Guinand (1964)
502:, specifically a form of
468:of the Fano plane is the
439:There are 64 choices for
101:in the projective plane.
946:Shioda, Tetsuji (1995),
60:complex projective plane
1002:Real algebraic geometry
909:, Berlin: Adolph Marcus
698:McKay & Sebbar 2007
606:Journal of Graph Theory
449:homogeneous coordinates
729:10.4153/cmb-1964-038-6
430:
349:
250:
43:
35:
942:. As cited by Cayley.
485:theta characteristics
431:
350:
251:
41:
33:
879:. pp. 373–386.
504:McKay correspondence
494:of genus 4, form a "
483:, and to the 28 odd
369:
296:
136:
52:quartic plane curve
836:Journal of Algebra
782:10.1007/BF03023483
743:, pp. 387–389
652:Arnold's trinities
514:, as discussed at
498:" in the sense of
451:of a point of the
426:
345:
339:
246:
245:
44:
36:
922: = 3",
916:Riemann, G. F. B.
894:978-3-540-30307-7
809:The Eightfold Way
628:10.1002/jgt.20597
481:del Pezzo surface
416:
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16:(Redirected from
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68:Euclidean plane
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961:(1): 109–128,
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842:(1): 457–486,
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723:(3): 399–404.
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659:on 2011-04-11
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577:Cayley (1879)
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474:Coxeter graph
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470:Heawood graph
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95:cubic surface
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849:math/0507118
839:
835:
808:
766:(2): 59–67,
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757:
754:Gray, Jeremy
746:
740:
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670:
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657:the original
651:
644:
609:
605:
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568:
561:Trott (1997)
556:
544:
531:
515:
492:sextic curve
489:
478:
462:
438:
357:
288:
281:
269:
258:
118:
114:
110:
108:
72:
64:real numbers
50:, a general
48:plane curves
45:
903:Plücker, J.
537:Gray (1982)
111:Trott curve
18:Trott curve
991:Categories
981:(1): 15–28
924:Ges. Werke
707:References
682:PostScript
466:Levi graph
453:Fano plane
284:dual curve
265:bitangents
127:polynomial
803:Reprinted
768:CiteSeerX
619:1002.1960
535:See e.g.
517:trinities
412:
173:−
129:equation
56:bitangent
932:citation
905:(1839),
868:17374533
798:14602496
700:, p. 11)
87:ellipses
967:1336422
827:1722415
790:0672918
612:: 1–9,
496:trinity
457:d, e, f
447:as the
445:a, b, c
272:M-curve
105:Example
77: (
75:Plücker
54:has 28
965:
891:
866:
825:
815:
796:
788:
770:
636:754481
634:
415:
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358:where
123:degree
951:(PDF)
864:S2CID
844:arXiv
794:S2CID
745:. In
632:S2CID
614:arXiv
524:Notes
510:and E
261:genus
125:four
83:locus
938:link
889:ISBN
813:ISBN
455:and
282:The
109:The
79:1839
881:doi
854:doi
840:304
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725:doi
686:PDF
678:TeX
624:doi
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