Real projective plane

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In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. 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 following diagram:

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Formal construction

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. Three self-intersecting embeddings (immersions) are Boy's surface, the Roman surface, and a sphere with a cross-cap.

See also