Göttingen Collection of Mathematical Models and Instruments

Complex surface according to Klein

Model 95

Felix KleinA VII 9-35


Complex surface according to Klein. Lead on wood.


Similar to model 124 by Rohn, this model depicts the Kummer surface with 16 real singularities, which is the maximal possible number for algebraic surfaces of degree 4.

Kummer surfaces are projective zero sets of quartic polynomials of the form: \[(x^2+y^2+z^2-\mu w^2)^2-\lambda pqrs\]where \( \lambda = \frac{3\mu-1}{3-\mu} \) and the equations \( p=0,q=0,r=0,s=0 \) define planes which correspond to the faces of a tetrahedron: \begin{align*} p&=w-z-\sqrt{2} x, & q&=w-z+\sqrt{2} x, \\ r&=w+z+\sqrt{2}y, & s&=w+z-\sqrt{2} y \end{align*}

In order to obtain this particular model, one has to choose \( \mu \approx 1.5 \) and consider the affine map \(w=1\).

Text written by: Malte Heuer and Thorsten Groth

Translated by: Lea Renner

Showcase of this model is Case number 1


Separataband M3 im Mathematischen Institut p. 557.

Separataband M3 im Mathematischen Institut p. 715.

Dyck, Walter(1892). Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente, Hof- und Universitätsdruckerei von Dr.C.Wolf und Sohn, München, Zur Ausstellung bei der Versammlung der Deutschen Mathematiker-Vereinigung 1893 in München, p. 283f..