Fano plane
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In finite geometry, the Fano plane is the projective plane with the least number of points and lines: 7 each.
According to the general construction (Method 2) explained in projective plane 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:
- points: 1,2,3,4,5,6,7
- lines: {1,2,4},{2,3,5},{3,4,6},{4,5,7},{5,6,1},{6,7,2},{7,1,3}
Using the standard construction via homogeneous coordinates, we can also identify the points by 3 binary digits, but not 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 bits modulo 2 for each position. Thus the 7 points correspond to the 7 non-identity elements of the group Z2 × Z2 × Z2 = Dih2 × Z2. The lines connecting three points correspond to the group operation (a, b, and c on one line means a+b=c, a+c=b, and b+c=a) or, correspondingly, to the Z2 × Z2 subgroups. The automorphism group of the group is that of the Fano plane, and of order 168.
On 3 lines the codes for the points have the 0 in a specific position (001 010 011, 001 100 101, 010 100 110), on 3 lines the codes for the points 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 1 (011 101 110).
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: any ordered pair is isomorphic 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. One out of every 30 permutations of the 7 points is an isomorphy, 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 symmetry group is made up of 6 conjugacy classes:
- 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 3rd point on AB, D to 3rd point on BC, etc.
- 24 with A mapped to B, B to C, C to 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 (i.e. with no 3 consecutive vertices on one line): 8 (21)
- non-degenerate pentagon (i.e. with no 3 consecutive vertices on one line): 2 (84)
- non-degenerate hexagon (i.e. with no 3 consecutive vertices on one line): 6 (28)
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.
The group is isomorphic to the group PSL(2,7) = PSL(3,2), and general linear group GL(3,2). It can be visualized as a group of permutations of symmetric partitions of the eightfold cube.
