Real Projective Plane

The real projective plane is the space obtained by taking every line through the origin in three-dimensional space as a single point, so that it can also be described as the ordinary two-dimensional sphere with every pair of exactly opposite points glued together into one point. It is the simplest example of a non-orientable surface with no boundary, meaning that a two-dimensional figure moved all the way around certain loops on the surface returns to its starting point as its own mirror image, a property it shares with the more easily visualized Mobius strip, which appears as a piece cut out of the real projective plane. Unlike the Mobius strip, the real projective plane cannot be embedded in ordinary three-dimensional space without the surface passing through itself, so physical and computer-generated models of it, such as the well known Boy's surface described by the German mathematician Werner Boy in 1901, always involve some form of self-intersection. The real projective plane also arises naturally in projective geometry as the completion of the ordinary Euclidean plane by a single extra line at infinity, added so that any two distinct lines, including previously parallel ones, always meet in exactly one point.