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# Serie IX, Nr 4

## Surface of order four. The surface consists of ten parts, which are connected in
twelve *A*_{1} 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=4/3 and λ=1/2.
In this case, the sphere cuts each edge of the tetrahedron twice.
This is a surface in the four dimensional space, given by the equation:
(x^{2} + y^{2} + z^{2} - 4/3 k^{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*=50 mm.
It consists of four congruent parts, touching each other in the double points.
For general values of &lambda>0, this gives twelve ordinary double points *A*_{1}.
**Title model and translation**

Fläche vierter Ordnung. Die Fläche besteht aus zehn Teilen, die in zwölf conischen
Knotenpunkten zusammenhängen.
Surface of order four. The surface consists of ten parts, which are connected in twelve
*A*_{1} 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 26,50

**Dimension**

11-11 cm

**Material**

Plaster

**Literature**

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