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Fano plane

In finite geometry, the Fano plane (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 PG(2, 2). Here, PG 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).

Fano plane

2

VII.2

23 × 3 × 7
PGL(3, 2)

7

7

Desarguesian
Self-dual

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.[1] "The" Fano plane of 7 points and lines is "a" Fano plane.

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.

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.

Group-theoretic construction[edit]

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

The identity permutation

21 permutations with two 2-cycles

42 permutations with a 4-cycle and a 2-cycle

56 permutations with two 3-cycles

There are 7 points with 24 symmetries fixing any point and dually, there are 7 lines with 24 symmetries fixing any line. The number of symmetries follows from the 2-transitivity of the collineation group, which implies the group acts transitively on the points.

There are 42 of points, and each may be mapped by a symmetry onto any other ordered pair. For any ordered pair there are 4 symmetries fixing it. Correspondingly, there are 21 unordered pairs of points, each of which may be mapped by a symmetry onto any other unordered pair. For any unordered pair there are 8 symmetries fixing it.

ordered pairs

There are 21 consisting of a line and a point on that line. Each flag corresponds to the unordered pair of the other two points on the same line. For each flag, 8 different symmetries keep it fixed.

flags

There are 7 ways of selecting a of four (unordered) points no three of which are collinear. These four points form the complement of a line, which is the diagonal line of the quadrangle and a collineation fixes the quadrangle if and only if it fixes the diagonal line. Thus, there are 24 symmetries that fix any such quadrangle. The dual configuration is a quadrilateral consisting of four lines no three of which meet at a point and their six points of intersection, it is the complement of a point in the Fano plane.

quadrangle

There are = 35 triples of points, seven of which are collinear triples, leaving 28 non-collinear triples or . The configuration consisting of the three points of a triangle and the three lines joining pairs of these points is represented by a 6-cycle in the Heawood graph. A color-preserving automorphism of the Heawood graph that fixes each vertex of a 6-cycle must be the identity automorphism.[2] This means that there are 168 labeled triangles fixed only by the identity collineation and only six collineations that stabilize an unlabeled triangle, one for each permutation of the points. These 28 triangles may be viewed as corresponding to the 28 bitangents of a quartic.[10] There are 84 ways of specifying a triangle together with one distinguished point on that triangle and two symmetries fixing this configuration. The dual of the triangle configuration is also a triangle.

triangles

There are 28 ways of selecting a point and a line that are not incident to each other (an anti-flag), and six ways of permuting the Fano plane while keeping an anti-flag fixed. For every non-incident point-line pair (p, l), the three points that are unequal to p and that do not belong to l form a triangle, and for every triangle there is a unique way of grouping the remaining four points into an anti-flag.

There are 28 ways of specifying a in which no three consecutive vertices lie on a line, and six symmetries fixing any such hexagon.

hexagon

There are 84 ways of specifying a in which no three consecutive vertices lie on a line, and two symmetries fixing any pentagon.

pentagon

The Fano plane contains the following numbers of configurations of points and lines of different types. For each type of configuration, the number of copies of configuration multiplied by the number of symmetries of the plane that keep the configuration unchanged is equal to 168, the size of the entire collineation group, provided each copy can be mapped to any other copy (see Orbit-stabiliser theorem). Since the Fano plane is self-dual, these configurations come in dual pairs and it can be shown that the number of collineations fixing a configuration equals the number of collineations that fix its dual configuration.


The Fano plane is an example of an (n3)-configuration, that is, a set of n points and n lines with three points on each line and three lines through each point. The Fano plane, a (73)-configuration, is unique and is the smallest such configuration.[11] According to a theorem by Steinitz[12] configurations of this type can be realized in the Euclidean plane having at most one curved line (all other lines lying on Euclidean lines).[13]

Each point is contained in 7 lines and 7 planes.

Each line is contained in 3 planes and contains 3 points.

Each plane contains 7 points and 7 lines.

Each plane is to the Fano plane.

isomorphic

Every pair of distinct planes intersect in a line.

A line and a plane not containing the line intersect in exactly one point.

The Fano plane can be extended in a third dimension to form a three-dimensional projective space, denoted by PG(3, 2). It has 15 points, 35 lines, and 15 planes and is the smallest three-dimensional projective space.[16] It also has the following properties:[17]

Projective configuration

Transylvania lottery

"Fano Plane". MathWorld.

Weisstein, Eric W.