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