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# Serie VII, Nr 10

## Surface of order three with an *A*_{2} double point, whose tangent
planes at that point intersect the surface in three lines. The model is also valid for the transfer of
the *A*_{2} into a *D*_{4} double point.

**Mathematical description**

The cubic surfaces are given by a homogeneous polynomial of degree 3. If the surface is smooth, it
contains 27 lines. In case the surface has isolated singularities (or double points), some of the lines
collapse. There is a Coxeter diagram associated with every double point. The singularities can be of
type *A*_{1}, *A*_{2}, *A*_{3}, *A*_{4},
*A*_{5}, *D*_{4}, *D*_{5} and *E*_{6}. There are
only 20 possible combinations.

This cubic surface has one *A*_{2} double point and 15 lines. Its tangent planes at that
point cut the surface along 3 real lines.

**Title model and translation**

Fläche dritter Ordnung mit *B*_{3}, dessen Ebenen in je drei reellen Knotenstrahlen
schneiden. Das model dient gleichzeitig zur überführung des *B*_{3} in einen
*U*_{6}.

Surface of order three with an *A*_{2} double point, whose tangent planes at that point
intersect the surface in three lines. The model is also valid for the transfer of the
*A*_{2} into a *D*_{4} double point.

**Text on Sticker and translation**

Wrong sticker.

**Designer**

Dr. Carl Rodenberg

Darmstadt in 1881

**Company**

L. Brill in Darmstadt

**Original price**

Mark 10,-

**Dimension**

10-15 cm

**Material**

Plaster

**Literature**

Schilling, M., Catalog mathematischer Modelle für den höheren
mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911