Thursday,  May 5

    Some final remarks on the projective plane and visualizing Flatland and Space.:

    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!
    In the projective plane.... Yes- the parallel line meets the line at a point on the horizon.

    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, and "Colin's world"- the hyperbolic plane.