Fano plane Totally Explained

Everything about Fano Plane totally explained

In finite geometry, the Fano plane (after Gino Fano) is the projective plane with the least number of points and lines: 7 each.

Geometry

Perhaps the best way to view the plane is via linear algebra. Using the standard construction via homogeneous coordinates, we can identify the points with the non-zero ordered triples of binary digits, for example, 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. 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, for example the collinear triples, correspond to the subgroups of order 4, for example, 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.
According to the general construction (Method 2) explained in the article on projective planes we've (with a slightly more compact notation) points P, 0, 1, 00, 01, 10, 11 and the following lines:
ť One line L = 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 doesn't correspond to interesting geometry but it can be interesting for coding 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. 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. 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).
   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:

the identity,

21 point permutations of type (12)(34) that keep all 3 points on one line fixed, and for one of these points, the other 2 lines through it; they interchange the other 4 points pairwise, and the other 4 lines ditto,

56 point permutations of type (123)(456) that rotate one triangle (a cyclic permutation of the 3 vertices, and a corresponding cyclic permutation of the 3 other points on the sides, keeping the 7th point fixed; hence "rotations about a point"); in other words: keep one point fixed, and choose 3 other points on a line, carry out a cyclic permutation of the 3 points on the line, and a corresponding cyclic permutation of the 3 other points.

42 point permutations of type (12)(3456) that keep one point fixed, interchange the other two points on one line through the fixed point, and perform a cyclic permutation of the remaining 4.

two classes of point permutations of type (1234567) :

24 with A mapped to B, B to C, C to the 3rd point on AB, D to 3rd point on BC, etc.
24 with A mapped to B, B to C, C to the 3rd point on AC, D to 3rd point on BD, etc.

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

point: 24 (7)

line: 24 (7)

set of two points: 8 (21)

figure consisting of two points of different color: 4 (for two given colors there are 42)

triangle: 6 (28)

triangle with 2 vertices of one given color and one of a different given color: 2 (84)

triangle with 3 vertices of different given colors: 1 (168)

non-degenerate quadrangle (for example with no 3 consecutive vertices on one line): 8 (21)

non-degenerate pentagon (for example with no 3 consecutive vertices on one line): 2 (84)

non-degenerate hexagon (for example with no 3 consecutive vertices on one line): 6 (28) In each case, up to isomorphism there's 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. 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's a valuable example in (block) design theory.
Matroid theory
ť Main article: Matroid theory

The Fano plane is one of the important examples in the structure theory of matroids. Excluding the Fano plane as a minor is necessary to characterize several important classes of matroids, such as regular, graphic, and cographic ones.

Further Information

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