Nonorientable Surface -- from Wolfram MathWorld
- ️Weisstein, Eric W.
A surface such as the Möbius strip or Klein bottle (Gray 1997, pp. 322-323) on which there exists a closed path such that the directrix is reversed when moved around this path. 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
and restricting 
to 
and 
to ![[0,pi/2]](http://mathworld.wolfram.com/images/equations/NonorientableSurface/Inline16.svg)
defines a map of the real projective plane
to 
.
In three dimensions, there is no unbounded nonorientable surface which does not intersect itself (Kuiper 1961, Pinkall 1986).
See also
Boy Surface, Cross-Cap, Klein Bottle, Möbius Strip, Orientable Surface, Real Projective Plane, Roman Surface
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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. 14 in Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, pp. 317-340, 1997.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.
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Cite this as:
Weisstein, Eric W. "Nonorientable Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NonorientableSurface.html