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In finite geometry, the Fano plane (after Gino Fano) is the projective plane with the least number of points and lines: 7 each. A finite geometry is any geometric system that has only a finite number of points. Gino Fano ( 5 january 1871 - 8 november 1952) was an Italian Mathematician. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics

Contents

Geometry

Perhaps the best way to view the plane is via linear algebra. Linear algebra is the branch of Mathematics concerned with Using the standard construction via homogeneous coordinates, we can identify the points with the non-zero ordered triples of binary digits, i. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations e. , excluding 000. This can be done in such a way that for every two points we can find the third point on the line through the two by adding modulo 2 in each position. In other words, the points of the Fano plane correspond to the non-zero points of the finite vector space F23 of dimension 3 over F2 , the finite field of order 2. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A line in the Fano plane corresponds to a 2-dimensional subspace of F23: the points a, b, c are collinear if and only if a + b = c (equivalently, b + c = a, or c + a = b).

This might be a bit simpler if we ignore the field structure of F23. Then the 7 points of the plane correspond to the 7 non-identity elements of the group (Z2)3 = Z2 × Z2 × Z2. The lines, i. e. the collinear triples, correspond to the subgroups of order 4, i. e. , those isomorphic to Z2 × Z2. The automorphism group of the group (Z2)3 is that of the Fano plane (see below), and has order 168. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

According to the general construction (Method 2) explained in the article on projective planes we have (with a slightly more compact notation) points P, 0, 1, 00, 01, 10, 11 and the following lines:

One line L = { P, 0, 1}
2 lines L0 = {P, 00, 10}, L1 = {P, 01, 11}
4 lines L00 = {0, 00, 01}, L01 = {1, 00, 11}, L10 = {0, 10, 11}, L11 = {1, 10, 01}

An alternative naming is:

The lines can be classified into four types. On 3 lines the codes for the points have the 0 in a constant position (001 010 011, 001 100 101, 010 100 110). On 3 lines the vectors have equal bits in two specific positions (001 110 111, 010 101 111, 100 011 111), and on one line the codes for the points all have exactly two bits equal to 1 (011 101 110). (This classification does not correspond to interesting geometry but it can be interesting for coding theory. Coding theory is one of the most important and direct applications of Information theory. )

Automorphism group and configurations

A permutation of the seven points that carries collinear points (points on the same line) to collinear points (in other words, it "preserves collinearity") is called a "collineation", "automorphism", or "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 A collineation is a one-to-one map from one Projective space to another or from a Projective plane onto itself such that the images of collinear points are themselves In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself 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 collineation group (or automorphism group, or symmetry group) is of order 168: any ordered pair is automorphic to any other one, and in addition to choosing to which ordered pair one ordered pair is mapped, we can choose the image of one more point, not on the same line, so we get 7 × 6 × 4 = 168 possibilities. A collineation is a one-to-one map from one Projective space to another or from a Projective plane onto itself such that the images of collinear points are themselves In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself 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 other words, there are 168 ordered triples forming a triangle (28 triangles, with for each 6 permutations of the vertices), all isomorphic, and the image of one determines the images of the other 4 points.

The collineation group is isomorphic to the projective special linear group PSL(2,7) = PSL(3,2), and the general linear group GL(3,2) (which are equal because the field has only one nonzero element). In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group 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

One out of every 30 permutations of the 7 points is an automorphism, so if we consider colorings of the 7 points of the Fano plane in 7 different given colors, up to isomorphism 30 different ones exist.

The automorphism group is made up of 6 conjugacy classes, which we describe in terms of their permutations of the points:

Order of symmetry groups of figures with (in parentheses) the number of them (the product is 168)

In each case, up to isomorphism there is only one (in the case of colors: for given colors).

In the three cases of the triangle, if we take the large one in the figure, the symmetry group corresponds to that of Euclidean symmetry of the figure.

Block design theory

The Fano plane is a small symmetric block design, specifically a (7,3,1)-design. In combinatorial Mathematics, a symmetric design is a Block design with equal numbers of points and blocks In combinatorial Mathematics, a block design (more fully a balanced incomplete block design) is a particular kind of Set system, which has long-standing The points of the design are the points of the plane, and the blocks of the design are the lines of the plane. As such it is a valuable example in (block) design theory.

Matroid theory

Main article: Matroid theory

The Fano plane is one of the important examples in the structure theory of matroids. In Combinatorics, a branch of Mathematics, a matroid ( or independence structure is a structure that captures the essence of a notion of "independence" In Combinatorics, a branch of Mathematics, a matroid ( or independence structure is a structure that captures the essence of a notion of "independence" Excluding the Fano plane as a minor is necessary to characterize several important classes of matroids, such as regular, graphic, and cographic ones. In Combinatorics, a branch of Mathematics, a matroid ( or independence structure is a structure that captures the essence of a notion of "independence"

References

See also

PlanetMath is a free, collaborative online Mathematics Encyclopedia. In combinatorial Mathematics, an incidence structure is a triple C=(PLI Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. In Mathematics, specifically Projective geometry, a configuration consists of a finite set of points and a finite set of lines such that each point is incident to
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