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# Serie IX, Nr 5

## Surface of order four. The surface consists of six congruent parts, which are
connected in four *D*_{4} 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 (of degree 2):
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=3 and λ=1/2.
In this model the sphere meets the vertices of the tetrahedron.
This is a surface in the four dimensional space, given by the equation:
(x^{2} + y^{2} + z^{2} - 3k^{2})^{2} =
1/2 [(z - k)^{2} - 2x^{2}] [(z + k)^{2} - 2y^{2}]
The model shows the same surface in the third dimension by taking *k*=25 mm.
It consists of four congruent parts, touching each other in the double points.
For general values of λ, this gives four *D*_{4} double points.
**Title model and translation**

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

Surface of order four. The surface consists of six congruent parts, which are connected in four
*D*_{4} double points.

**Text on Sticker and translation**

Fl. 4. Ord. mit 4 dobbelebenen.

Surface of order four with four double planes.

**Designer**

Dr. Kummer

Berlin in 1883

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 24,50

**Dimension**

10-10 cm

**Material**

Plaster

**Literature**

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