A finite geometry is any geometric system that has only a finite number of points. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact precisely the same number of points as there are real numbers. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Mathematics, the real numbers may be described informally in several different ways A finite geometry can have any (finite) number of dimensions.
Finite planes
The following remarks apply only to finite planes. There are two kinds of finite plane geometry: affine and projective. Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. In an affine geometry, the normal sense of parallel lines applies. Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths In a projective plane, by contrast, any two lines intersect at a unique point, and so parallel lines do not exist. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject
An affine plane geometry is a nonempty set X (whose elements are called "points"), along with a nonempty collection L of subsets of X (whose elements are called "lines"), such that:
- Given any two distinct points, there is exactly one line that contains both points.
- The parallel postulate: Given a line
and a point p not on , there exists exactly one line 
containing p such that 
or 
. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive - There exists a set of four points, no three of which belong to the same line.
The last axiom ensures that the geometry is not empty, while the first two specify the nature of the geometry. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members
The simplest affine plane contains only four points; it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel". More generally, a finite affine plane of order n has n2 points and n2 + n lines; each line contains n points, and each point is on n + 1 lines.
A projective plane geometry is a nonempty set X (whose elements are called "points"), along with a nonempty collection L of subsets of X (whose elements are called "lines"), such that:
- Given any two distinct points, there is exactly one line that contains both points.
- The intersection of any two distinct lines contains exactly one point.
- There exists a set of four points, no three of which belong to the same line.
An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each This suggests the principle of duality for projective plane geometry, meaning that any true statement about the geometry remains true if we exchange points for lines and lines for points. While the third axiom only requires the existence of four points, the plane must contain at least seven points in order to satisfy the first two axioms. In this simplest of the projective planes, there are also seven lines; each point is on three lines, and each line contains three points. This particular projective plane is sometimes called the Fano plane. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each 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. For this reason, the Fano plane is called the projective plane of order 2. In general, the projective plane of order n has n2 + n + 1 points and the same number of lines (respecting duality); each line contains n + 1 points, and each point is on n + 1 lines.
A permutation of the Fano plane's seven points that carries collinear points (points on the same line) to collinear points is called a "symmetry" of the plane. In Geometry, the Relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L' and 'intersects' (as in 'line L1 Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The full symmetry group is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2), and general linear group GL(3,2). The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, the Projective special linear group PSL(27 is a finite Simple group that has important applications in Algebra, In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation It is well-established that both affine and projective planes of order n exist when n is a prime power, a prime number raised to a positive integer exponent. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French It is conjectured that no finite planes exist with orders that are not prime powers, although this statement has not been proved. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of The best result to date is the Bruck-Ryser theorem, which states: If n is a positive integer of the form 4k + 1 or 4k + 2 and n is not equal to the sum of two integer squares, then n does not occur as the order of a finite plane. The Bruck – Chowla – Ryser theorem is a result on the Combinatorics of Block designs It states that if a ( v, b A negative number is a Number that is less than zero, such as −2 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Algebra, the square of a number is that number multiplied by itself The smallest integer that is not a prime power and not covered by the Bruck-Ryser theorem is 10; 10 is of the form 4k + 2, but it is equal to the sum of squares 12 + 32. Using sophisticated techniques and computer analysis, it has been shown that 10 is also not the order of a finite plane. The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.
Finite spaces of 3 or more dimensions
For some important differences between finite plane geometry and the geometry of higher-dimensional finite spaces, see axiomatic projective space. In Mathematics, a finite projective space S is a set P (the set of points together with a set of subsets of P (the set of lines all of which For a discussion of higher-dimensional finite spaces in general, see, for instance, the works of J.W.P. Hirschfeld.