Nonorientable Surface

A surface such as the Möbius Strip on which there exists a closed path such that the directrix is reversed when moved around this path. The Euler Characteristic of a nonorientable surface is $\leq 0$. The real Projective Plane is also a nonorientable surface, as are the Boy Surface, Cross-Cap, and Roman Surface, all of which are homeomorphic to the Real Projective Plane (Pinkall 1986). There is a general method for constructing nonorientable surfaces which proceeds as follows (Banchoff 1984, Pinkall 1986). Choose three Homogeneous Polynomials of Positive Even degree and consider the Map

$${\bf f}=(f_1(x,y,z), f_2(x,y,z), f_3(x,y,z)):\Bbb{R}^3\to \Bbb{R}^3.$$ (1)

Then restricting $x$, $y$, and $z$ to the surface of a sphere by writing

$$x = \cos\theta\sin\phi$$ (2)

$$y = \sin\theta\sin\phi$$ (3)

$$z = \cos\phi$$ (4)

and restricting $\theta$ to $[0, 2\pi)$ and $\phi$ to $[0, \pi/2]$ defines a map of the Real Projective Plane to $\Bbb{R}^3$.

In 3-D, there is no unbounded nonorientable surface which does not intersect itself (Kuiper 1961, Pinkall 1986).

See also Boy Surface, Cross-Cap, Möbius Strip, Orientable Surface, Projective Plane, Roman Surface

References

Banchoff, T. ``Differential Geometry and Computer Graphics.'' In Perspectives of Mathematics: Anniversary of Oberwolfach (Ed. W. Jager, R. Remmert, and J. Moser). Basel, Switzerland: Birkhäuser, 1984.

Gray, A. ``Nonorientable Surfaces.'' Ch. 12 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 229-249, 1993.

Kuiper, N. H. ``Convex Immersion of Closed Surfaces in $E^3$.'' Comment. Math. Helv. 35, 85-92, 1961.

Pinkall, U. ``Models of the Real Projective Plane.'' Ch. 6 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 63-67, 1986.