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 l in Euclid's geometry is there a unique  line through P that is parallel to l? Euclid's answer... YES!
Question:Can that be proven from a list of properties (axioms) about the plane???

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

and inversion?
Other "pseudo-flat" worlds are possible- though if one lives in such a world, the world would look like flatland close by.   Such worlds include the plane (Flatland) , the torus, the projective plane, ....

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."