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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.
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 ...
Military use of combat shotguns through the 20th century has created a need for ammunition maximizing the combat effectiveness of such weapons within the limitations of international law. 12-gauge has been widely accepted as an appropriate bore diameter to provide an effective number of projectiles within an acceptable recoil. Early 12-gauge ...
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.
If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called the projective plane of order 2 because it is unique (up to isomorphism). In general, the projective plane of order n has n 2 + n + 1 points and the same number of lines; each line ...
A Hughes plane H: [1] is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1, has a Desarguesian Baer subplane H 0, is a self-dual plane in which every orthogonal polarity of H 0 can be extended to a polarity of H, every central collineation of H 0 extends to a central collineation of H, and
Although it may be embedded in two dimensions, the Desargues configuration has a very simple construction in three dimensions: for any configuration of five planes in general position in Euclidean space, the ten points where three planes meet and the ten lines formed by the intersection of two of the planes together form an instance of the configuration. [2]
A set of n − 1 MOLS of order n can be used to construct a projective plane of order n (and conversely). A ( v , k , λ) difference set is a subset D of a group G such that the order of G is v , the size of D is k , and every nonidentity element of G can be expressed as a product d 1 d 2 −1 of elements of D in exactly λ ways (when G is ...