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 orthogonal circles in wingeometry?
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?
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.
[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
The Topological Classification of "closed surfaces."