real projective plane

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 Mbius 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:

     ---->     ^   |     |   v     <----

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 two-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.

Real Projective Plane

Formal construction

See also