Clebsch's diagonal surface
Clebsch's diagonal surface is a cubic surface with no double points. Such a surface contains exactly 27 lines. On the diagonal surface all 27 lines are real.Photos of 3D prints
The Clebsch diagonal surface is a generic but highly symmetric degree-3 surface. It contains 27 lines, all of which are real. They are coloured so as to display Schläfli's „double six“ (model 401) configuration: 6 red lines and 6 blue lines so that every line touches five lines of the opposite colour. In projective coordinates, the diagonal surface may be given by the equation $(x+y+z+w)^3=x^3+y^3+z^3+w^3.$ This model was designed by Prof. C. Rodenberg.
Showcase of this model is Case number 3
Schilling, Martin(Hrg.): Catalog mathematischer Modelle, Leipzig(Verlag von Martin Schilling) 1911, 7.Auflage, No.44. p. 116.
Fischer, Gerd(Hrg.): Fotoband: Mathematische Modelle / Mathematical Models, mit 132 Fotografien, Braunschweig/Wiesbaden (Vieweg) 1986, picture(s) 10-12. .
Fischer, Gerd(Hrg.): Mathematical Models, Commentary, Braunschweig/Wiesbaden(Vieweg) 1986. .
Rodenberg. Modelle von Flächen 3. Ordnung . Online version
Schläfli(1858). An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface. The Quarterly Journal of Pure and Applied Mathematics II, p. 55-65; 110-120. Online version