If we negate it, we get a version of non-Euclidean geometry. There are two options: Parallel Postulate for Spherical Geometry. For a given line g and a point P not on g, there are no lines through P parallel to g. Parallel Postulate for Hyperbolic Geometry. For a given line g and a point P not on g, there is more than one line through P parallel to g.

Before string theory introduced the concept of extra dimensions, the fascination with strange warping of space in the 1800s was perhaps nowhere as clear as in the creation of non-Euclidean geometry, where mathematicians began to explore new types of geometry that weren’t based on the rules laid out 2,000 years earlier by Euclid. One version of non-Euclidean geometry is …

sorted out a key concept in geometry. He made a general study of curvature of spaces in all dimensions. In 2-dimensions: Euclidean geometry is flat (curvature = 0) and any triangle angle sum = 180 degrees. The non-Euclidean geometry of Lobachevsky is negatively curved, and any triangle angle sum < 180 degrees. The geometry of the sphere is positively