Projective Plane in a sentence

Some disciplines in mathematics restrict the meaning of projective plane to only this type of projective plane since otherwise general statements about projective spaces would always have to mention the exceptions when the geometric dimension is two.

The real (or complex) projective plane and the projective plane of order 3 given above are examples of Desarguesian projective planes.

Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structure C* is also a projective plane, called the dual plane of C. If C and C* are isomorphic, then C is called self-dual.

There is a projective plane of order N if and only if there is an affine plane of order N. When there is only one affine plane of order N there is only one projective plane of order N, but the converse is not true.

In this construction, each "point" of the real projective plane is the one-dimensional subspace through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a plane through the origin in the 3-space.

The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane.

Another longstanding open problem is whether there exist finite projective planes of prime order which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order).

Finite projective planes It can be shown that a projective plane has the same number of lines as it has points (infinite or finite).

Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus.

Fano subplanes A Fano subplane is a subplane isomorphic to PG(2,2), the unique projective plane of order 2. If you consider a quadrangle (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane.

Affine planes Projectivization of the Euclidean plane produced the real projective plane.

A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed.

A projective plane satisfying Pappus's theorem universally is called a Pappian plane.

Desargues' theorem and Desarguesian planes The theorem of Desargues is universally valid in a projective plane if and only if the plane can be constructed from a three-dimensional vector space over a skewfield as above.

Note however that the projective plane RP 2 is not the one-point compactification of the plane R 2 since more than one point is added.

Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or a degenerate plane.

Vector space construction Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case.

Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other.

Any connected sum involving a real projective plane is nonorientable.

Constant positive curvature: A Möbius band of constant positive curvature cannot be complete, since it is known that the only complete surfaces of constant positive curvature are the sphere and the projective plane.