# Clebsch's diagonal surface

### Model 135

### Description

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### Additions

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

### References

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