Fano plane - WikiHQ

thumb|The Fano plane

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. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is . Here, stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one).

The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study.

In a separate usage, a Fano plane is a projective plane that never satisfies Fano's axiom; in other words, the diagonal points of a complete quadrangle are always collinear. "The" Fano plane of 7 points and lines is "a" Fano plane.

A standard visualization of the Fano plane draws its seven points as the vertices, edge midpoints, and centroid of an equilateral triangle, and its seven lines as the three sides and three symmetry axes of the triangle, together with a circle through the three edge midpoints. However, the visual differences between the positions of these points and the shapes of these lines are merely an artifact of the visualization. As an abstract structure, the Fano plane is highly symmetric, with symmetries that take any point to any other point or that take any line to any other line.

Homogeneous coordinates

The Fano plane can be constructed via linear algebra as the projective plane over the finite field with two elements. One can similarly construct projective planes over any other finite field, with the Fano plane being the smallest.

Using the standard construction of projective spaces via homogeneous coordinates, the seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits 001, 010, 011, 100, 101, 110, and 111. This can be done in such a way that for every two points p and q, the third point on line pq has the label formed by adding the labels of p and q modulo 2 digit by digit (e.g., 010 and 111 resulting in 101). In other words, the points of the Fano plane correspond to the non-zero points of the finite vector space of dimension 3 over the finite field of order 2.

Due to this construction, the Fano plane is considered to be a Desarguesian plane, even though the plane is too small to contain a non-degenerate Desargues configuration (which requires 10 points and 10 lines).

The lines of the Fano plane may also be given homogeneous coordinates, again using non-zero triples of binary digits. With this system of coordinates, a point is incident to a line if the coordinate for the point and the coordinate for the line have an even number of positions at which they both have nonzero bits: for instance, the point 101 belongs to the line 111, because they have nonzero bits at two common positions. In terms of the underlying linear algebra, a point belongs to a line if the inner product of the vectors representing the point and line is zero.

The lines can be classified into three types.

On three of the lines the binary triples for the points have the 0 in a constant position: the line 100 (containing the points 001, 010, and 011) has 0 in the first position, and the lines 010 and 001 are formed in the same way.
On three of the lines, two of the positions in the binary triples of each point have the same value: in the line 110 (containing the points 001, 110, and 111) the first and second positions are always equal to each other, and the lines 101 and 011 are formed in the same way.
In the remaining line 111 (containing the points 011, 101, and 110), each binary triple has exactly two nonzero bits.

Group-theoretic construction

Alternatively, the 7 points of the plane correspond to the 7 non-identity elements of the group . The lines of the plane correspond to the subgroups of order 4, isomorphic to . The automorphism group GL(3, 2) of the group (Z2)3 is that of the Fano plane, and has order 168.

Levi graph

thumb|Bipartite Heawood graph. Points are represented by vertices of one color and lines by vertices of the other color.

As with any incidence structure, the Levi graph of the Fano plane is a bipartite graph, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are incident. This particular graph is a connected cubic graph (regular of degree 3), has girth 6 and each part contains 7 vertices. It is the Heawood graph, the unique 6-cage.

Collineations

180px|thumb|left|A collineation of the Fano plane corresponding to the 3-bit [[Gray code permutation]]

A collineation, automorphism, or symmetry of the Fano plane is a permutation of the 7 points that preserves collinearity: that is, it carries collinear points (on the same line) to collinear points. By the Fundamental theorem of projective geometry, the full collineation group (or automorphism group, or symmetry group) is the projective linear group ,

This is a well-known group of order 168 = 23·3·7, the next non-abelian simple group after A5 of order 60 (ordered by size).

As a permutation group acting on the 7 points of the plane, the collineation group is doubly transitive meaning that any ordered pair of points can be mapped by at least one collineation to any other ordered pair of points. (See below.)

Collineations may also be viewed as the color-preserving automorphisms of the Heawood graph (see figure).

Dualities

A bijection between the point set and the line set that preserves incidence is called a duality and a duality of order two is called a polarity.

Dualities can be viewed in the context of the Heawood graph as color reversing automorphisms. An example of a polarity is given by reflection through a vertical line that bisects the Heawood graph representation given on the right. The existence of this polarity shows that the Fano plane is self-dual. This is also an immediate consequence of the symmetry between points and lines in the definition of the incidence relation in terms of homogeneous coordinates, as detailed in an earlier section.

Cycle structure

thumb|180px|A [[nimber numbering of the Fano plane]]

The permutation group of the 7 points has 6 conjugacy classes.

These four cycle structures each define a single conjugacy class:

40px The identity permutation
40px 21 permutations with two 2-cycles
40px 42 permutations with a 4-cycle and a 2-cycle
40px 56 permutations with two 3-cycles

The 48 permutations with a complete 7-cycle form two distinct conjugacy classes with 24 elements:

40px A maps to B, B to C, C to D. Then D is on the same line as A and B.
40px A maps to B, B to C, C to D. Then D is on the same line as A and C.

(See here for a complete list.)

The number of inequivalent colorings of the Fano plane with <math>n</math> colors can be calculated by plugging the numbers of cycle structures into the Pólya enumeration theorem. This number of colorings is