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 . 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

(1) |

(2) | |||

(3) | |||

(4) |

and restricting to and to defines a map of the Real Projective Plane to .

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

**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 .'' *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.

© 1996-9

1999-05-25