A popular idea in popular books on physics and math is to illustrate the failure of Euclidean geometry on the surface of a sphere. Parallel lines do intersect! Well, that’s being loose with definitions and examples. Meridian lines of longitude are tangentially parallel at the equator and converge at the poles. Why are those considered to be “parallel” lines? Why pick only lines that are great circles? Instead, take two lines that are non-great-circles that are the intersection of parallel planes( in the 3-space the sphere is imbedded in) with the surface of the sphere? They never intersect. In the 2-space surface of the sphere, the lines have constant separation, throughout their continuous length. That postulate has no trouble in non-Euclidean geometry if one takes the most advantageous definitions.
Spheres and parallel lines
Category: Physics