## 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 . 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)

Then restricting , , and to the surface of a sphere by writing
 (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).

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 .'' 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.