- Real projective plane
In

mathematics , the**real**is the space of lines inprojective plane **R**^{3}passing through the origin. It is a non-orientable two-dimensionalmanifold , that is, asurface , that has basic applications togeometry , but which cannot be embedded in our usual three-dimensional space without intersecting itself. It hasEuler characteristic of 1 giving a genus of 1.The real projective plane is sometimes described in terms of a construction based on the

Möbius strip : if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane.Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together, the real projective plane can thus be represented as a unit square ( [0,1] × [0,1] ) with its sides identified by the following

equivalence relation s::(0, "y") ~ (1, 1 − "y") for 0 ≤ "y" ≤ 1

and

:("x", 0) ~ (1 − "x", 1) for 0 ≤ "x" ≤ 1,

as in the diagram on the right.

**Construction**Consider a

sphere , and let thegreat circle s of the sphere be "lines", and let pairs ofantipodal point s be "points". It is easy to check that it obeys the axioms required of aprojective plane :*any pair of distinct great circles meet at a pair of antipodal points;

*and any two distinct pairs of antipodal points lie on a single great circle.This is the

**real projective plane**.If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points.

The projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y

iff y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in**R**^{3}.The resulting surface, a 2-dimensional compact non-orientable manifold, is a little hard to visualize, because it cannot be embedded in 3-dimensional

Euclidean space without intersecting itself.The quotient map from the sphere onto the real projective plane is in fact a (two-to-one)

covering map . It follows that thefundamental group of the real projective plane is the cyclic group of order 2, i.e. integers modulo 2. One can take the loop "AB" from the figure above to be the generator.**Immersing the real projective plane in three-space**The projective plane cannot be embedded (that is without intersection) in three-dimensional space. However, it can be immersed (local neighbourhoods do not have self-intersections).

Boy's surface is an example of an immersion.The

Roman surface is a more degenerate map of the projective plane into 3-space, containing across-cap . The same goes for a sphere with across-cap .The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assume that it does embed, it would bound a compact region in three-dimensional Euclidean space by the Generalized Jordan Curve Theorem. The outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but the boundary manifold would be

projective space , which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.A

polyhedral representation is thetetrahemihexahedron .Looking in the opposite direction, the hemi-cube,

hemi-dodecahedron , andhemi-icosahedron , abstract regular polychora, can be constructed as a regular figure in the "projective plane".**Homogeneous coordinates**The set of lines in the plane can be represented using

homogeneous coordinates . A line "ax"+"by"+"c"=0 can be represented as ("a":"b":"c"). These coordinates have the equivalence relation ("a":"b":"c") = ("da":"db":"dc") for all non zero values of "d". Hence a different representation of the same line "dax"+"dby"+"dc"=0 has the same coordinates. The set of coordinates ("a":"b":1) gives the usualreal plane , and the set of coordinates ("a":"b":0) defines aline at infinity .**Embedding into 4-dimensional space**The projective plane embeds into 4-dimensional Euclidean space. Consider $mathbb\; RP^2$ to be the quotient of the two-sphere $S^2\; =\; \{(x,y,z)\; in\; mathbb\; R^3\; :\; x^2+y^2+z^2\; =\; 1\}$ by the antipodal relation $(x,y,z)sim\; (-x,-y,-z),$. Consider the function $mathbb\; R^3\; o\; mathbb\; R^4$ given by $(x,y,z)longmapsto\; (xy,xz,y^2-z^2,2yz)$. This map restricts to a map whose domain is $S^2$ and, since it is a purely quadratic polynomial, it can be factorised to give a map $mathbb\; RP^2\; o\; mathbb\; R^4$. Moreover, this map is an embedding. Notice that this embedding admits a projection into $R^3$ which is the

Roman surface .**Higher genus**The article on the

fundamental polygon provides a description of the real projective planes of higher genus.**ee also***

Projective space

*Pu's inequality for real projective plane**External links***

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