Projective plane

two-dimensional projective space

An incidence structure $\pi = \{ {\mathcal P} , {\mathcal L} , I \}$. The elements of the set $ {\mathcal P} $ are called points, the elements of the set $ {\mathcal L} $ are called (straight) lines and $ I $ is an incidence relation. An incidence structure satisfies the following axioms:

1) for any two distinct points $ p $ and $ q $ there is a unique line $ L $ such that $ pIL $ and $ qIL $;

2) for any two distinct lines $ L $ and $ M $ there is a unique point $ p $ such that $ pIL $ and $ pIM $; and

3) there exist four points no three of which are incident with one line.

Figure: p075310a

For example, the set $ \Pi $ of lines and planes of three-dimensional affine space, passing through a point $ o $, is a projective plane if as projective points one takes the lines of $ \Pi $ and as projective lines one takes the planes of $ \Pi $. In this interpretation the homogeneous coordinates of a point of the projective plane over a field have a clear geometric meaning as the coordinates of some vector on the line corresponding to the point (cf. Projective geometry; Projective coordinates). Another example is the projective plane constituted by seven points $ A _ {i} $, $ i = 1 \dots 7 $, and the seven lines $ \{ A _ {1} , A _ {2} , A _ {4} \} $, $ \{ A _ {2} , A _ {3} , A _ {5} \} $, $ \{ A _ {3} , A _ {4} , A _ {6} \} $, $ \{ A _ {4} , A _ {5} , A _ {7} \} $, $ \{ A _ {5} , A _ {6} , A _ {1} \} $, $ \{ A _ {6} , A _ {7} , A _ {2} \} $, $ \{ A _ {7} , A _ {1} , A _ {3} \} $( Fig. a). It is a representative of the class of finite projective planes. A projective plane $ P ( 2, n) $ is called a finite projective plane of order $ n $ if the incidence relation satisfies one more axiom:

4) there is a line incident with exactly $ n + 1 $ points.

In $ P ( 2, n) $ every point (line) is incident with $ n + 1 $ lines (points), and the number of points of the plane, which is equal to the number of lines, is $ n ^ {2} + n + 1 $. The question for which values of $ n $ a projective plane $ P ( 2, n) $ exists is unanswered (1990). The existence of a finite projective plane whose order is a power of a prime number has been proved (cf. [4]). The non-existence of $ P ( 2, n) $ has been proved for a large class of numbers: If $ n $ is congruent to 1 or 2 modulo 4 and if in the prime factorization of $ n $ there is at least one prime number congruent to 3 modulo 4 that occurs with odd exponent, then $ P ( 2, n) $ does not exist. Such are, e.g., $ n = 6, 14, 21, 22 , . . . $. The problems remain open for $ n = 10, 12, 15, 18 , . . . $. An important problem in the theory of finite projective planes is the study of subplanes of a given $ P ( 2, n) $. E.g., if $ P ( 2, m) $ is a proper subplane of $ P ( 2, n) $, then $ m ^ {2} + m \leq n $ or $ m ^ {2} = n $( cf. [5]).

The concept of duality is characteristic for projective planes. Two projective planes are said to be dual if between the points (lines) of one plane and the lines (points) of the other there is a one-to-one correspondence preserving incidence. Certain projective planes (e.g. projective planes over a field) admit a dual mapping onto itself, called a polarity; projective planes admitting a polarity are called self-dual. The so-called little duality principle holds for projective planes: If some statement $ {\mathcal A} $ about points and lines of a projective plane formulated solely in terms of incidence between them holds, then the statement $ {\mathcal B} $ dual to $ {\mathcal A} $ also holds, i.e. the statement obtained from $ {\mathcal A} $ by replacing the word "point" with "line" and vice versa also holds.

An isomorphic mapping of a projective plane onto itself is called a collineation. A collineation of a finite projective plane $ P ( 2, n) $ is a permutation of the set of points and a permutation of the set of lines, where these two permutations are compatible. A finite projective plane is Desarguesian if it has a group of collineations that acts doubly-transitively on its points. The group of collineations of a Desarguesian projective plane $ \mathop{\rm PG} ( 2, p ^ {h} ) $ has order

$$ h ( p ^ {2h} + p ^ {h} + 1) ( p ^ {2h} + p ^ {h} ) p ^ {2h} ( p ^ {h} - 1) ^ {2} . $$

The group of collineations of a non-Desarguesian projective plane $ P ( 2, n) $ has order at most

$$ n ^ {s} ( n ^ {2} + n + 1) ( n ^ {2} + n) n ^ {2} ( n - 1) ^ {2} , $$

where $s \leq \log_{2} n$. The orders of the groups of collineations of the known non-Desarguesian projective planes do not exceed the orders of the groups of collineations of the Desarguesian planes of the same order.

The Lenz–Barlotti classification of projective planes is based on considering the 53 types of sets $$ T(G) = \{ (x,X) : G \textrm{is} (x, X) \textrm{transitive} \}, $$ defined for full collineation groups $G$. One of the basic approaches to study projective planes is the introduction of coordinates and a ternary operation in these planes. To every possible type of projective plane in the Lenz–Barlotti classification corresponds a system of algebraic laws that must be satisfied by the natural coordinate domain of the projective planes determined by the ternary operation. E.g., a projective plane is Desarguesian (Pappian) if and only if all its natural coordinate domains are skew-fields (fields). A Desarguesian finite projective plane $ P ( 2, n) $ is Pappian.

A special feature of a finite Desarguesian projective plane $ \mathop{\rm PG} ( 2, n) $ is that it has a collineation of order $ n ^ {2} + n + 1 $ that acts cyclically on points and lines. This makes it possible to represent $ \mathop{\rm PG} ( 2, n) $ by a cyclic array. This representation of $ \mathop{\rm PG} ( 2, n) $ is as follows. Its points, enumerated by the natural numbers from 1 to $ n ^ {2} + n + 1 $, are distributed over a rectangular array of $ n + 1 $ rows and $ n ^ {2} + n + 1 $ columns such that each column, which represents a line with all points on it, is obtained by adding 1 (modulo $ n ^ {2} + n + 1 $) to the elements of the previous column. E.g., a representation of $ P ( 2, 2) $ is

$$ \begin{array}{c} 1 \\ 2 \\ 4 \end{array} \ \begin{array}{c} 2 \\ 3 \\ 5 \end{array} \ \begin{array}{c} 3 \\ 4 \\ 6 \end{array} \ \begin{array}{c} 4 \\ 5 \\ 7 \end{array} \ \begin{array}{c} 5 \\ 6 \\ 1 \end{array} \ \begin{array}{c} 6 \\ 7 \\ 2 \end{array} \ \begin{array}{c} 7 \\ 1 \\ 3 \end{array} $$

The planes $ P ( 2, n) $, $ n \leq 8 $, are unique up to an isomorphism — they are Desarguesian or Galois planes (cf. [6]), while already for $ n = 9 $ four non-isomorphic planes are known (cf. [7]).

If to the axioms of a projective plane and Desargues' assumption one adds order axioms (described by separation of pairs of points lying on one straight line, e.g. in Fig. b the pair $ C, D $ separates the pair $ A, B $, while $ A, C $ does not separate $ B, D $) and the continuity axiom, then the projective plane thus obtained is isomorphic to the real affine plane completed by improper elements: an improper (infinitely-distant) point is added to each line, to parallel lines the same point, to non-parallel lines distinct points, and the improper points are required to ly on one improper straight line.

Figure: p075310b

A projective plane is called topological if the sets of its points and lines are topological spaces and if the join and intersection are continuous. In a topological plane the ternary operation is continuous in all its arguments. From the topological point of view the point set of a real projective plane (as well as the set of lines) is a closed non-orientable manifold whose Euler characteristic is 1.

References

[1]	H.S.M. Coxeter, "Projective geometry" , Cambridge Univ. Press (1987)
[2]	R. Baer, "Linear algebra and projective geometry" , Acad. Press (1972)
[3]	L.A. Skornyakov, "Projective planes" Uspekhi Mat. Nauk , 6 : 6 (1951) pp. 112–154 (In Russian)
[4]	F. Kárteszi, "Introduction to finite geometries" , North-Holland (1976)
[5]	M. Hall, "The theory of groups" , Macmillan (1959) pp. Chapt. 20
[6]	R. Dembowski, "Finite geometries" , Springer (1968) pp. 254
[7]	T.G. Room, P.B. Kirkpatrick, "Miniquaternion geometry" , Cambridge Univ. Press (1971)

Comments

A projective plane is called Desarguesian if the Desargues assumption holds in it (i.e. if it is isomorphic to a projective plane over a skew-field).

The idea of finite projective planes (and spaces) was introduced by K. von Staudt [a5], pp. 87–88.

The fact that a finite projective plane with doubly-transitively acting group of collineations is Desarguesian is the Ostrom–Wagner theorem.

A finite Desarguesian $ P( 2, n) $ is Pappian, since a finite skew-field is commutative.

A collineation of order $ n ^ {2} + n+ 1 $ of $ PG( 2, n) $ is called a Singer cycle, [a7]. It has now been settled by exhaustive computer search that there are no projective planes of order 10, and precisely four non-isomorphic projective planes of order 9. The algebraic structure coordinatizing a projective plane is usually called a planar ternary ring. The flag-transitive finite projective planes have been determined by W.M. Kantor [a1].

References

[a1]	W.M. Kantor, "Primitive permutation groups of odd degree, and an application to finite projective planes" J. Algebra , 106 (1987) pp. 15–45
[a2]	G. Pickert, "Projective Ebenen" , Springer (1975)
[a3]	D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1973)
[a4]	H. Lüneburg, "Translation planes" , Springer (1979)
[a5]	K.G.C. von Staudt, "Beiträge zur Geometrie der Lage" , I , Korn , Nürnberg (1865)
[a6]	G. Fano, "Sui postulati fondamentali della geometria proiettiva" Giornale di Mat. , 30 (1892) pp. 106–132
[a7]	I. Singer, "A theorem in finite projective geometry and some applications to number theory" Trans. Amer. Math. Soc. , 43 (1938) pp. 377–385

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