Projective
The real projective plane RP2 is the space of all lines in Euclidean three dimensional space R3 going through the origin. Equivalently, it is the quotient of a sphere by the antipodal map. RP2 is a non-orientable surface and cannot be mapped to three-dimensional space without self intersections. There are however three aesthetically interesting mapping into R3: Boy's surface, the cross-capped disk and the Roman surface.
Boy's surface
Boy's surface is an immersion of the projective plane into R3, which roughly means that although the map self-intersects, its tangent space never degenerates. Neither the cross-capped disk or the Roman surface are immersions.
Here is a stereographic image: look at the left image with your right eye, and vice versa, to see the image in three-dimensions.
Views from random directions.
The cross-capped disk
The Roman surface
