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The Projective Plane

The Projective Plane, which is abbreviated as RP^2, is the surface with Euler characteristic = 1. It is obtained by idendifying antipodal points on the boundary of a disk. It is closed and non-orientable, which implies that its image cannot be viewed in 3-dimensions without self-intersections. The projective plane may also be obtained by attatching a disk to a Möbius band: each has one boundary component along which to do the attatching.

There are many functions which carry RP^2 into R^3. Three famous surfaces which are the images of a function f: RP^2 -> R^3 are the Crosscap, the Roman surface, and Boy's surface.

Description and pictures of the Crosscap.

Description and pictures of the Roman surface.

Description and pictures of Boy's Surface.

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