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.

Schoutte (1910) showed that the 27 lines can be put into a One-to-One correspondence with the vertices of a particular Polytope in 6-D space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the Polytope and conversely (Du Val 1931). A similar correspondence can be made between the 28 bitangents of the general plane Quartic Curve and a 7-D Polytope (Coxeter 1928) and between the tritangent planes of the canonical curve of genus 4 and an 8-D 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).

**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.). *Mathematical Models from the Collections of Universities and Museums.*
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/~wwwagag/Galerie.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.

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 Surface of Third Order into Species.'' *Phil. Trans. Roy. Soc.* **153**, 193-247, 1864.

Schoutte, 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.

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1999-05-25