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In mathematics, the real projective plane is the space of lines in R³ passing through the origin. It is a non-orientable two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space without intersecting itself. It has Euler characteristic of 1 giving a genus of 1.

It is often described intuitively, in relation with a Möbius strip: it would result if one could glue the single edge of the strip to itself in the correct direction. Or in other words, a square [0,1] × [0,1] with sides identified by the relations:

(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.

Contents 1 Formal construction 2 Trying to embed the real projective plane in three-space 3 Homogeneous coordinates 4 Embedding into 4-dimensional space 5 Higher genus 6 See also

Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that it obeys the axioms required of a projective 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 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 projective plane cannot strictly 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 another interesting example, but this contains cross-caps so it is not technically an immersion. The same goes for a sphere with a cross-cap.

The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: If it did 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.

A polyhedral representation is the Tetrahemihexahedron.

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 usual real plane, and the set of coordinates (a:b:0) defines a line at infinity.

The projective plane does embed into 4-dimensional Euclidean space. Using homogeneous coordinates, the projective plane corresponds to points such that x² + y² + z² = 1 subject to the relation (x,y,z)˜( − x, − y, − z). The embedding into R⁴ is given by the function .

The article on the fundamental polygon provides a description of the real projective planes of higher genus.

• projective space