Thursday,  December 4

    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;

 Mobius Strip

    Experiments with the mobius band.
    a sphere,
    a torus


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

Klein Bottle

    the projective plane

Boy's Surface ;

    spheres with handles,
    spheres with cross caps
    Crosscap .

    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."