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A projective plane of order N is a Steiner S(2, N + 1, N 2 + 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.
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]
If P is a finite set, the projective plane is referred to as a finite projective plane. The order of a finite projective plane is n = k – 1, that is, one less than the number of points on a line. All known projective planes have orders that are prime powers. A projective plane of order n is an ((n 2 + n + 1) n + 1) configuration. The smallest ...
Hanfried Lenz gave a classification scheme for projective planes in 1954, [6] which was refined by Adriano Barlotti in 1957. [7] This classification scheme is based on the types of point–line transitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes.
The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2.
The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t = 2 or (recently) t ≥ 2.
As an example, when λ = 1 and b = v, we have a projective plane: X is the point set of the plane and the blocks are the lines. A symmetric balanced incomplete block design or SBIBD is a BIBD in which v = b (the number of points equals the number of blocks). They are the single most important and well studied subclass of BIBDs.
According to a theorem by Steinitz [12] configurations of this type can be realized in the Euclidean plane having at most one curved line (all other lines lying on Euclidean lines). [13] The upper figure is an alternative representation of the Fano plane in grid layout – compare with one of the finite projective plane of order 3 below