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Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point.

Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane.[1] This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP2, or P2(R), among other notations. 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, not shared by all projective planes.

Examples[edit]

The extended Euclidean plane[edit]

To turn the ordinary Euclidean plane into a projective plane proceed as follows:

2 : all isomorphic to PG(2, 2)

3 : all isomorphic to PG(2, 3)

4 : all isomorphic to PG(2, 4)

5 : all isomorphic to PG(2, 5)

6 : impossible as the order of a projective plane, proved by who showed that Euler's thirty-six officers problem has no solution. However, the connection between these problems was not known until Bose proved it in 1938.[13]

Tarry

7 : all isomorphic to PG(2, 7)

8 : all isomorphic to PG(2, 8)

9 : PG(2, 9), and three more different (non-isomorphic) : a Hughes plane, a Hall plane, and the dual of this Hall plane. All are described in (Room & Kirkpatrick 1971).

non-Desarguesian planes

10 : impossible as an order of a projective plane, proved by heavy computer calculation.

[14]

11 : at least PG(2, 11), others are not known but possible.

12 : it is conjectured to be impossible as an order of a projective plane.

It can be shown that a projective plane has the same number of lines as it has points (infinite or finite). Thus, for every finite projective plane there is an integer N ≥ 2 such that the plane has


The number N is called the order of the projective plane.


The projective plane of order 2 is called the Fano plane. See also the article on finite geometry.


Using the vector space construction with finite fields there exists a projective plane of order N = pn, for each prime power pn. In fact, for all known finite projective planes, the order N is a prime power.


The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is the Bruck–Ryser–Chowla theorem that if the order N is congruent to 1 or 2 mod 4, it must be the sum of two squares. This rules out N = 6. The next case N = 10 has been ruled out by massive computer calculations. Nothing more is known; in particular, the question of whether there exists a finite projective plane of order N = 12 is still open.


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).


A projective plane of order N is a Steiner S(2, N + 1, N2 + N + 1) system (see Steiner system). Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes.


The number of mutually orthogonal Latin squares of order N is at most N − 1. N − 1 exist if and only if there is a projective plane of order N.


While the classification of all projective planes is far from complete, results are known for small orders:

Projective planes in higher-dimensional projective spaces[edit]

Projective planes may be thought of as projective geometries of "geometric" dimension two.[15] Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane. These turn out to be "tamer" than the projective planes since the extra degrees of freedom permit Desargues' theorem to be proved geometrically in the higher-dimensional geometry. This means that the coordinate "ring" associated to the geometry must be a division ring (skewfield) K, and the projective geometry is isomorphic to the one constructed from the vector space Kd+1, i.e. PG(dK). As in the construction given earlier, the points of the d-dimensional projective space PG(dK) are the lines through the origin in Kd+1 and a line in PG(dK) corresponds to a plane through the origin in Kd+1. In fact, each i-dimensional object in PG(dK), with i < d, is an (i + 1)-dimensional (algebraic) vector subspace of Kd+1 ("goes through the origin"). The projective spaces in turn generalize to the Grassmannian spaces.


It can be shown that if Desargues' theorem holds in a projective space of dimension greater than two, then it must also hold in all planes that are contained in that space. Since there are projective planes in which Desargues' theorem fails (non-Desarguesian planes), these planes can not be embedded in a higher-dimensional projective space. Only the planes from the vector space construction PG(2, K) can appear in projective spaces of higher dimension. 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.[16]

– a generalization of a finite projective plane.

Block design

Combinatorial design

Difference set

Incidence structure

Generalized polygon

Projective geometry

Non-Desarguesian plane

Smooth projective plane

Transversals in finite projective planes

– a projective plane with one vertex removed.

Truncated projective plane

VC dimension of a finite projective plane

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