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The Fano plane is the projective plane with the fewest points and lines. The smallest 2-dimensional projective geometry (that with the fewest points) is the Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
The affine planes which arise from the projective planes PG(2, q) are denoted by AG(2, q). 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. The affine planes formed by the removal ...
C ∗ is also a projective plane, called the dual plane of C. If C and C ∗ are isomorphic, then C is called self-dual. The projective planes PG(2, K) for any field (or, more generally, for every division ring (skewfield) isomorphic to its dual) K are self-dual. In particular, Desarguesian planes of finite order are always self-dual.
In the Cayley plane, lines and points may be defined in a natural way so that it becomes a 2-dimensional projective space, that is, a projective plane. It is a non-Desarguesian plane, where Desargues' theorem does not hold. More precisely, as of 2005, there are two objects called Cayley planes, namely the real and the complex Cayley plane.
A two-dimensional complex space – such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface – has two complex dimensions, which can alternately be represented using four real dimensions. A two-dimensional lattice is an infinite grid of points which can be represented using integer coordinates.
There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space . Not all projective planes can be embedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known as Desargues' theorem ...
One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass through a particular origin point; in this model, lines through the origin are considered to be the "points" of the projective plane, and planes through the origin are considered to be the "lines" in the projective plane.
The Fano plane, the projective plane over the field with two elements, is one of the simplest objects in Galois geometry.. Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). [1]