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Cubic Surface

An algebraic surface of order 3. Schläfli and Cayley classified the singular cubic surfaces. On the general cubic, there exists a curious geometrical structure called double sixes, and also a particular arrangement of 27 (possibly complex) lines, as discovered by Schläfli (Salmon 1965, Fischer 1986) and sometimes called Solomon's seal lines. A nonregular cubic surface can contain 3, 7, 15, or 27 real lines (Segre 1942, Le Lionnais 1983). The Clebsch diagonal cubic contains all possible 27. The maximum number of ordinary double points on a cubic surface is four, and the unique cubic surface having four ordinary double points is the Cayley cubic.

Examples of cubic surfaces include the Cayley cubic, Clebsch diagonal cubic, ding-dong surface, handkerchief surface, Möbius strip, monkey saddle, shoe surface, Wallis's conical edge, and Whitney umbrella.

Schoute (1910) showed that the 27 lines can be put into a one-to-one correspondence with the vertices of a particular polytope in six-dimensional space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the polytope and conversely (Du Val 1933). A similar correspondence can be made between the 28 bitangents of the general plane quartic curve and a seven-dimensional polytope (Coxeter 1928) and between the tritangent planes of the canonical curve of genus 4 and an eight-dimensional polytope (Du Val 1933).

A smooth cubic surface contains 45 tritangents (Hunt). The Hessian of smooth cubic surface contains at least 10 ordinary double points, although the Hessian of the Cayley cubic contains 14 (Hunt).

Cayley Cubic, Clebsch Diagonal Cubic, Ding-Dong Surface, Double Sixes, Eckardt Point, Isolated Singularity, Solomon's Seal Lines, Tritangent

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References

Bruce, J. and Wall, C. T. C. "On the Classification of Cubic Surfaces." J. London Math. Soc. 19, 245-256, 1979.Cayley, A. "A Memoir on Cubic Surfaces." Phil. Trans. Roy. Soc. 159, 231-326, 1869.Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc. 24, 7-9, 1928.Du Val, P. "On the Directrices of a Set of Points in a Plane." Proc. London Math. Soc. Ser. 2 35, 23-74, 1933.Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, pp. 9-14, 1986.Fladt, K. and Baur, A. Analytische Geometrie spezieler Flächen und Raumkurven. Braunschweig, Germany: Vieweg, pp. 248-255, 1975.Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni-kl.de/~hunt/drawings.html.Hunt, B. "The 27 Lines on a Cubic Surface" and "Cubic Surfaces." Ch. 4 and Appendix B.4 in The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 108-167 and 302-310, 1996.Klein, F. "Über Flächen dritter Ordnung." Gesammelte Abhandlungen, Band II. Berlin: Springer-Verlag, pp. 11-62, 1973.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 49, 1983.Rodenberg, C. "Zur Classification der Flächen dritter Ordnung." Math. Ann. 14, 46-110, 1878.Salmon, G. Analytic Geometry of Three Dimensions. New York: Chelsea, 1965.Schläfli, L. "On the Distribution of Surfaces of Third Order into Species, in Reference to the Absence or Presence of Singular Points, and the Reality of Their Lines." Philos. Trans. Roy. Soc. London 153, 193-241, 1863.Schoute, P. H. "On the Relation Between the Vertices of a Definite Sixdimensional Polytope and the Lines of a Cubic Surface." Proc. Roy. Acad. Amsterdam 13, 375-383, 1910.Segre, B. The Nonsingular Cubic Surface. Oxford, England: Clarendon Press, 1942.

Cubic Surface

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Weisstein, Eric W. "Cubic Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubicSurface.html