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 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 1931). 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).