**Curves and Surfaces:
More about proof and what is possible? what is not?**

**A very old problem: In Euclid's geometry,
there are lines that never meet.... but is this true about Flatland?
How can someone in Flatland tell whether 2 lines are parallel?
Question:Given a point P and a line **

*Show video:"A
non-euclidean Universe."
*Show orthogonal circles in wingeometry?

**and inversion?
**

** Surfaces:**Mark Sudduth's
web page of surfaces.[ A physics master's degree student at UT, Arlington.

**What is a surface?
Bounded, unbounded:
Closed, open:**

With or without boundary:

Orientable or Non-orientable:

Can be realized (imbedded) in a plane, in 3 space, in 4 space.

Can be visualized (immersed) in ...

**Examples:A closed disc, an open disc, a
plane, an annulus- cylinder, a mobius band;
**

**Experiments with the mobius band.
a sphere,
a torus**

** [Activity:Graphs on the torus?]
a Klein bottle**

** the projective plane
**

** spheres with handles,
spheres with cross caps **

**Visualizations of surfaces by flattened
- cut apart models.
A cylinder, a mobius band, the torus, the Klein bottle, the projective plane.
**

**Handles and cross-caps **
**attached to
the sphere.
The Topological Classification of "closed surfaces."
**