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A finite plane of order n is one such that each line has n points (for an affine plane), or such that each line has n + 1 points (for a projective plane). One major open question in finite geometry is: Is the order of a finite plane always a prime power? This is conjectured to be true.
In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point.
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 ...
If the number of points in an affine plane is finite, then if one line of the plane contains n points then: each line contains n points, each point is contained in n + 1 lines, there are n 2 points in all, and; there is a total of n 2 + n lines. The number n is called the order of the affine plane.
In incidence geometry, most authors [16] give a treatment that embraces the Fano plane PG(2, 2) as the smallest finite projective plane. An axiom system that achieves this is as follows: (P1) Any two distinct points lie on a line that is unique. (P2) Any two distinct lines meet at a point that is unique.
Initially discovered by Veblen and Wedderburn, this plane was generalized to an infinite family of planes by Marshall Hall. Hall planes are a subclass of the more general André planes. The dual of the Hall plane of order 9. Numerous other constructions of both finite and infinite non-Desarguesian planes are known, see for example Dembowski (1968).
A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. The order of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes).