info prev up next book cdrom email home

Solomon's Seal Lines

The 27 Real or Imaginary straight Lines which lie on the general Cubic Surface and the 45 triple tangent Planes to the surface. All are related to the 28 Bitangents of the general Quartic Curve.

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 and a 7-D Polytope (Coxeter 1928) and between the tritangent planes of the canonical curve of genus four and an 8-D Polytope (Du Val 1933).

See also Brianchon's Theorem, Cubic Surface, Double Sixes, Pascal's Theorem, Quartic Surface, Steiner Set


Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, pp. 322-325, 1945.

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.

Schoutte, P. H. ``On the Relation Between the Vertices of a Definite Sixdimensional Polytope and the Lines of a Cubic Surface.'' Proc. Roy. Akad. Acad. Amsterdam 13, 375-383, 1910.

© 1996-9 Eric W. Weisstein