1. The real projective plane can
be constructed
as a topological surface, by attaching a Mobius strip along its
circular
edge to the circular edge of a disk. Another construction of the
real
projective plane is to identify antipodal (diametrically opposite)
points
on a sphere. There is no way to represent this surface in three
dimensions
without the surface intersecting itself.
2. Steiner's Roman surface is one
representation of the real projective plane, and it intersects itself
along three line
segments.
3. These three line segments meet
each other at a triple point, and their six endpoints are called pinch
points.
4. Another interesting property of
Steiner's Roman surface is that at each of its points, there are
infinitely many conic section curves which go through that point and
lie on the surface.
5. The Roman surface can be defined
by parametric functions which are quotients of quadratic polynomials in
three variables:
Notice that the equations are "homogeneous,"
which means
that for any nonzero constant c,F(r,s,t)=F(cr,cs,ct).
Any surface defined by homogeneous quadratic rational
functions like this is called a "Steiner surface," and the Roman
surface is one of 10
types, as classified by Coffman, Schwartz, and Stanton.
6. The homogeneous property of the
parametric equations means that we don't have to use all three domain
variables (r,s,t), but can use just two parameters to describe the
surface. One twoparameter equation for the Roman surface
is:
7. Another twoparameter equation
for the
Roman surface is to use points (r,s,t) on the unit sphere, which itself
has
parametric equations
Since r^2+s^2+t^2=1, the composition of the
Roman map F and the Sphere map S, or F(S(u,v)), is
Since F(r,s,t)=F(r,s,t), the antipodal points
on the
sphere have the same image.

Steiner's first major publication
( Systematische Entwicklung der Abhängigkeit geometrischer
Gestalten von einander . . ., part 1, Berlin 322pp., 1832) laid
the foundation for modern synthetic geometry. Very soon after
this volume appeared numerous honors were bestowed on him. They
included an honorary degree from the University of Königsberg
(1833) and a new chair of geometry at the University of Berlin.
His Roman Surface papers appeared near the peak of an outstanding
career.
