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# Serie IX, Nr 1

## Surface of order four. The surface consists of four congruent parts, which are
connected in six *A*_{3} double points.

**Mathematical description**

This model belongs to the Kummer's series of surfaces with tetrahedral symmetry.
Kummer studied surfaces of degree 4 in complex projective space, given by:

φ^{2}=λpqrs,
where φ=0 is the equation of a sphere:
φ = (x^{2} + y^{2} + z^{2} - μk^{2})
and *p*,*q*,*r* and *s* are lineair equations:
p = z - k + x√(2),
q = z - k - x√(2),
r = -z - k + y√(2),
s = -z - k - y√(2).

The equation *pqrs*=0 is a regular tetrahedron concentric with the sphere.
The four sides of the tetrahedron have equations *p*=0,*q*=0,*r*=0 and *s*=0.
The parameter μ is the radius of the sphere φ=0.
When the sphere meets the edges of the tetrahedron, singularities arise.
For μ=1 there are no real intersection points.

The model represents the case then μ=1 and λ=-1/8.
Then, the sphere touches the six edges of the tetrahedron at their midpoints.
This is a surface in the four dimensional space, given by the equation:
(x^{2} + y^{2} + z^{2} - k^{2})^{2} =
-1/3 [(z - k)^{2} - 2x^{2}][(z + k)^{2} - 2y^{2}]
The model shows the same surface in the third dimension by taking *k*=40 mm.
It consist of four congruent parts, touching each other in the double points.
For general values of &lambda<0, this gives six double points *A*_{3}.
**Title model and translation**

Fläche vierter Ordnung. Die Fläche besteht aus vier congruenten Teilen, die in sechs
biplanaren Knotenpunkten zusammenhängen.

Surface of order four. The surface consists of four congruent parts, which are connected in six
*A*_{3} double points.

**Text on Sticker and translation**

Fl. 4 Ord. mit 4 doppelebenen.

Surface of order four with four double planes.

**Designer**

Dr. Kummer

Berlin in 1883

**Company**

L. Brill in Darmstadt

**Original price**

Mark 19,50

**Dimension**

11-11 cm

**Material**

Plaster

**Literature**

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