Finite geometry

A finite geometry is any geometric system that has only a finite number of points.

Related Topics:
Geometric - Finite - Points

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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.

Related Topics:
Euclidean geometry - Real number

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There are two main kinds of finite geometry: affine and projective.

Related Topics:
Affine - Projective

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In an affine geometry, the parallel postulate holds, meaning that the normal sense of parallel lines applies.

Related Topics:
Affine geometry - Parallel postulate - Parallel

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In a projective geometry, by contrast, any two lines intersect at a unique point, and so parallel lines do not exist.

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Both finite affine geometry and finite projective geometry may be described by fairly simple axioms.

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For affine geometry, the axioms are as follows:

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• Given any two distinct points, there is exactly one line that includes both points.
• The parallel postulate: Given a line L and a point P not on L, there exists exactly one line through P that is parallel to L.
• There exists a set of four points, no three collinear.

The last axiom ensures that the geometry is not empty, while the first two specify the nature of the geometry.

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The simplest affine plane contains only four points; it is called the affine plane of order 2.

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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".

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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.

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(Figures of affine planes of orders 2 and 3 to be added.)

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The axioms of projective geometry are:

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• Two distinct points lie on exactly one line.
• Two distinct lines intersect at exactly one point.
• There exists a set of four points, no three collinear.

Diagram of the Fano plane

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An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged.

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This establishes the principle of duality for projective geometry, meaning that any true statement about the geometry remains true if we exchange points for lines and lines for points.

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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.

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In this simplest of the projective planes, there are also seven lines; each point is on three lines, and each line contains three points.

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This particular projective plane is sometimes called the Fano plane.

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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.

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For this reason, the Fano plane is called the projective plane of order 2.

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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.

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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. 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). For a different representation of the Fano plane that allows study of this full group of 168 symmetries, see The Eightfold Cube.

Related Topics:
Collinear - Symmetry - Symmetry group - PSL(2,7) - General linear group - Group

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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.

Related Topics:
Prime number - Positive - Integer - Exponent

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It is conjectured that no finite planes exist with orders that are not prime powers, although this statement has not been proved.

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The best result to date is the Bruck-Ryser theorem, which states:

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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.

Related Topics:
Positive - Integer - Square

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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.

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Using sophisticated techniques and computer analysis, it has been shown that 10 is also not the order of a finite plane.

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The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.

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