The Real Projective Plane is a non-orientable surface obtained by identifying opposite sides of the rim of a hemisphere.  It is not immersible into R3 without self intersection.  The immersion shown above is obtained by connecting opposite sides of a hemisphere by a strip with a mobius twist.  I generated the full top as a locus of arcs who's vertical height varies as 1 + 0.5*sin(v)^2 where v is the angle around the circumference of the rim.  The top is an example of what is called a "cross cap".  Unfortunately this immersion is not regular as it has two pinch point singularities.  A regular immersion was discovered by Werner Boy in 1901 and is called Boy's Surface.