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# Serie IX, Nr 3

## Surface of order four. Steiner's Roman surface.

**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 special case λ=μ=1.
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} =
[(z-k)^{2}-2x^{2}][(z+k)^{2}-2y^{2}]
The surface was discovered by J. Steiner in Rome, 1844, and is called the Steiner surface or the Roman
surface. Steiner's Roman surface is one representation of the real projective plane.
The model shows this surface in the third dimension by taking *k*=50 mm.
This is a special form of Serie IX, nr 1, were the six double points
change in three double lines (also called pinch points or Whitney singularities) and meet in a triple
point of the surface.
The model consists of four congruent parts, touching each other in the double points.
For general values of λ, this gives six double points *A*_{3}.
**Title model and translation**

Fläche vierter Ordnung. Die römische Fläche von Steiner.

Surface of order four. Steiner's Roman surface.

**Text on Sticker and translation**

Fläche 4. Ord. von Steiner.

Surface of order four; Steiner's surface.

**Designer**

Dr. Kummer

Berlin in 1883

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 21,50

**Dimension**

10-10 cm

**Material**

Plaster

**Literature**

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