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Real Projective Plane


RealProjectivePlaneSquare

The real projective plane is the closed topological manifold, denoted RP^2, that is obtained by projecting the points of a plane E from a fixed point P (not on the plane), with the addition of the line at infinity. It can be described by connecting the sides of a square in the orientations illustrated above (Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324).

There is then a one-to-one correspondence between points in E and lines through P not parallel to E. Lines through P that are parallel to E have a one-to-one correspondence with points on the line at infinity. Since each line through P intersects the sphere S^2 centered at P and tangent to E in two antipodal points, RP^2 can be described as a quotient space of S^2 by identifying any two such points. The real projective plane is a nonorientable surface. The equator of S^2 (which, in the quotient space, is itself a projective line) corresponds to the line at infinity.

RealProjectivePlaneK6

The complete graph on 6 vertices K_6 can be drawn in the projective plane without any lines crossing, as illustrated above. Here, the projective plane is shown as a dashed circle, where lines continue on the opposite side of the circle. The dual of K_6 on the projective plane is the Petersen graph.

The Boy surface, cross-cap, and Roman surface are all homeomorphic to the real projective plane and, because RP^2 is nonorientable, these surfaces contain self-intersections (Kuiper 1961, Pinkall 1986).


See also

Boy Surface, Complex Projective Plane, Cross-Cap, Cross Surface, Henneberg's Minimal Surface, Nonorientable Surface, Projective Plane, Real Projective Space, Roman Surface

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References

Apéry, F. Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces. Braunschweig, Germany: Vieweg, 1987.Coxeter, H. S. M. The Real Projective Plane, 3rd ed. Cambridge, England: Cambridge University Press, 1993.Gardner, M. Martin Gardner's Sixth Book of Mathematical Games from Scientific American. New York: Scribner's, 1971.Geometry Center. "The Projective Plane." http://www.geom.umn.edu/zoo/toptype/pplane/.Gray, A. "Realizations of the Real Projective Plane." §14.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 330-335, 1997.Klein, F. §1.2 in Vorlesungen über nicht-euklidische Geometrie. New York: Springer-Verlag, 1968.Kuiper, N. H. "Convex Immersion of Closed Surfaces in E^3." Comment. Math. Helv. 35, 85-92, 1961.Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 64-65, 1986.

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Real Projective Plane

Cite this as:

Weisstein, Eric W. "Real Projective Plane." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RealProjectivePlane.html

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