In the theory of algebraicplane curves, a generalquartic plane curve has 28bitangent lines, lines that are tangent to the curve in two places. These lines exist in thecomplex projective plane, but it is possible to define quartic curves for which all 28 of these lines havereal numbers as their coordinates and therefore belong to theEuclidean plane.

An explicit quartic with twenty-eight real bitangents was first given byPlücker (1839)^[1] 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 thelocus of centers ofellipses with fixed axis lengths, tangent to two non-parallel lines.^[2]Shioda (1995) gave a different construction of a quartic with twenty-eight bitangents, formed by projecting acubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is theline at infinity in the projective plane.

TheTrott curve, another curve with 28 real bitangents, is the set of points (x,y) satisfying thedegree fourpolynomial equation

${\displaystyle {\displaystyle 144(x^4+y^4)-225(x^2+y^2)+350x^2{y}^2+81=0.}}$

These points form a nonsingular quartic curve that hasgenus three and that has twenty-eight realbitangents.^[3]

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 anM-curve. 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.

Thedual curve to a quartic curve has 28 real ordinary double points, dual to the 28 bitangents of the primal curve.

The 28 bitangents of a quartic may also be placed in correspondence with symbols of the form

wherea, b, c, d, e, f are all zero or one and where

${\displaystyle ad+be+cf=1(\mathrm{mod}2).}$^[4]

There are 64 choices fora, b, c, d, e, f, but only 28 of these choices produce an odd sum. One may also interpreta, b, c as thehomogeneous coordinates of a point of theFano plane andd, e, f 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.

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.^[5] TheLevi graph of the Fano plane is theHeawood graph, 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 theCoxeter graph.^[6]

The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2del Pezzo surface,^[5] and to the 28 oddtheta characteristics.

The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a canonicsextic curve of genus 4, form a "trinity" in the sense ofVladimir Arnold, specifically a form ofMcKay correspondence,^[7][8][9] and can be related to many further objects, including E₇ and E₈, as discussed attrinities.

• Blum, R.; Guinand, A. P. (1964). "A quartic with 28 real bitangents".Canadian Mathematical Bulletin.7 (3):399–404.doi:10.4153/cmb-1964-038-6.
• Cayley, Arthur (1879), "On the bitangents of a quartic",Salmon's Higher Plane Curves, pp. 387–389. InThe collected mathematical papers of Arthur Cayley, Andrew Russell Forsyth, ed., The University Press, 1896, vol. 11, pp. 221–223.
• Gray, Jeremy (1982), "From the history of a simple group",The Mathematical Intelligencer,4 (2):59–67,CiteSeerX 10.1.1.163.2944,doi:10.1007/BF03023483,MR 0672918,S2CID 14602496.Reprinted inLevy, Silvio, ed. (1999),The Eightfold Way, MSRI Publications, vol. 35, Cambridge University Press, pp. 115–131,ISBN 0-521-66066-1,MR 1722415.
• Manivel, L. (2006), "Configurations of lines and models of Lie algebras",Journal of Algebra,304 (1):457–486,arXiv:math/0507118,doi:10.1016/j.jalgebra.2006.04.029,S2CID 17374533.
• McKay, John; Sebbar, Abdellah (2007). "Replicable Functions: An Introduction".Frontiers in Number Theory, Physics, and Geometry II. pp. 373–386.doi:10.1007/978-3-540-30308-4_10.ISBN 978-3-540-30307-7.
• Plücker, J. (1839),Theorie der algebraischen Curven: gegrundet auf eine neue Behandlungsweise der analytischen Geometrie, Berlin: Adolph Marcus.
• Riemann, G. F. B. (1876), "Zur Theorie der Abel'schen Funktionen für den Fallp = 3",Ges. Werke, Leipzig, pp. 456–472{{citation}}: CS1 maint: location missing publisher (link). As cited by Cayley.
• Shioda, Tetsuji (1995),"Weierstrass transformations and cubic surfaces"(PDF),Commentarii Mathematici Universitatis Sancti Pauli,44 (1):109–128,MR 1336422
• Trott, Michael (1997), "Applying GroebnerBasis to Three Problems in Geometry",Mathematica in Education and Research,6 (1):15–28.

• Quartic curves
• Real algebraic geometry

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