Gluing a disk to this figure gives the real projective plane.
An infinite version of this shape is the only non-trivial line bundle of the circle. Gluing a disk to this figure gives the real projective plane. Repeatedly dividing up this shape will give rise to paradromic rings. Like the torus, this surface, which is commonly pictured embedded in three-dimensional space, has Euler characteristic zero. It is the simplest (*) non-orientable surface. Two of these shapes are formed by cutting a Klein bottle in half. This shape is often used in conveyor belts, because it allows them to have uniform wear without ever having to be flipped inside out. For 10 points, identify this strange surface with only one side, which can be formed by taping together the ends of a paper rectangle after giving it a twist. 
An infinite version of this shape is the only non-trivial line bundle of the circle.
Mobius strip [MÃ¶bius strip]