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 Poincare disc model for the hyperbolic plane.