The Fano plane is a finite projective plane of order 2, having the smallest possible number of points and lines (7 points and 7 lines), with three points on each line and with three lines passing through each point. Named after the Italian mathematician Gino Fano .
Homogeneous coordinates
The Fano plane can be constructed using linear algebra as a projective plane over a finite field with two elements. It is possible to construct projective planes over any other finite field in the same way, but the Fano plane will be the smallest.
Using the standard construction of projective spaces using homogeneous coordinates , seven points of the Fano plane can be marked with seven nonzero triples of binary digits 001, 010, 011, 100, 101, 110 and 111. For any pair of points p and q, the third point on the line pq has a label, obtained from the marks p and q by addition modulo 2; e.g. 110 + 011 = 101. In other words, the points of the Fano plane correspond to nonzero points of a finite vector space of dimension 3 over a finite field of order 2.
According to this construction, the Fano plane is considered to be desargue, although the plane is too small to contain a non - degenerate Desargues configuration (10 points and 10 lines are required).
Homogeneous coordinates can also be assigned to Fano lines, again using nonzero triples of binary digits. In this system, a point is incident to a line if the coordinates of the point and the coordinates of the line have an even number of positions in which both coordinates are nonzero bits. For example, point 101 belongs to line 111, because both the line and the point have non-zero bits in two common positions. In terms of linear algebra, a point belongs to a line if the scalar product of vectors representing the point and the line is zero.
Straight lines can be divided into three types.
- On three straight lines, binary codes for points have 0 in a constant position. So, on line 100 (containing points 001, 010 and 011), all points have 0 in the first position. Lines 010 and 001 have the same property.
- On three lines, the binary code of points has the same value in two positions. So, on line 110 (containing points 001, 110 and 111), the values of the first and second positions (coordinates) of the points are always the same. Lines 101 and 011 have a similar property.
- On the remaining line 111 (containing points 011, 101, and 110), each code has exactly two nonzero bits.
Symmetries
The permutation of the seven points of the Fano plane, preserving the incidence of points (line), that is, when a point lying on a line appears on the same line, is called “collineation”, “ automorphism ” or “ symmetry ” of the plane. The complete collineation group (either the automorphism group or the symmetry group ) is the projective linear group PGL (3,2) [1] , which in this case is isomorphic to the projective special linear group PSL (2,7) = PSL (3,2) and the full linear group GL (3,2) (which is equal to PGL (3,2), since the field has only one nonzero element). The group consists of 168 different permutations.
The automorphism group consists of 6 conjugacy classes .
All , with the exception of a cycle of length 7, uniquely determine the conjugacy class:
- The identity permutation.
- 21 permutation of two 2-cycles | 2-cycles]].
- 42 permutations of 4-cycles and 2-cycles.
- 56 permutations of 3-cycles.
48 permutations with a full cycle of length 7 form two conjugacy classes of 24 elements each:
- A goes to B , B to C , C to D. In this case, D lies on one line with A and B.
- A goes to B , B to C , C to D. In this case, D lies on one line with A and C.
Due to the Redfield - Poilly theorem, the number of nonequivalent colorings of the Fano plane in n colors is equal to:
Configurations
The Fano plane contains the following various configurations of points and lines. For each type of configuration, the number of copies of the configuration, multiplied by the number of symmetries of the plane at which the configuration is saved, is 168, the size of the entire group of symmetries.
- There are 7 points and 24 symmetries that preserve these points.
- There are 7 lines and 24 symmetries that preserve these lines.
- There are 7 options for choosing a quadrangle from four (unordered) points, none of which lie on one straight line and 24 symmetries that preserve such a quadrilateral. These four points form the complement of the line, which is the diagonal of the quadrilateral.
- There are 21 points, each of which can be converted by symmetry into any other disordered pair. For each disordered pair, there are 8 symmetries that preserve it.
- There are 21 flags consisting of a line and a point on it. Each flag corresponds to an unordered pair of other points lying on the same line. For each flag, there are 8 different symmetries that preserve it.
- There are 28 triangles that correspond to one-to-one 28 [2] . For each triangle, there are six symmetries that preserve it, one for each permutation of the points inside the triangle.
- There are 28 ways to select a point and a line that are not incident to each other ( antiflag ) and six ways to rearrange the Fano plane that preserve antiflag. For any pair of non-incident points and a line ( p , l ), three points that are not equal to p and do not belong to l form a triangle, and for any triangle there is only one way to group the remaining four points into antiflag.
- There are 28 ways to construct a hexagon in which no three consecutive vertices lie on one straight line, and six symmetries that preserve any such hexagon.
- There are 42 ordered pairs of points and again, each can be converted by symmetry into any other ordered pair. For ordered pairs, there are 4 symmetries that preserve it.
- There are 42 ways to select a quadrangle from four cyclically ordered points, none of which lie on one straight line, and four symmetries that preserve any such quadrilateral. For any non-oriented four there are two cyclic orders.
- There are 84 ways to select a triangle with a point on that triangle, and for each choice there are two symmetries that preserve this choice.
- There are 84 ways to select a pentagon , in which no three consecutive vertices lie on one straight line, and two symmetries that preserve any pentagon.
- There are 168 different ways of choosing a triangle with ordering its three vertices and only one identical symmetry that preserves this configuration.
Group
7 points of the plane correspond to 7 nonidentity elements of the group ( Z 2 ) 3 = Z 2 × Z 2 × Z 2 . Straight planes correspond to subgroups of order 4 isomorphic to Z 2 × Z 2 . The automorphism group group ( Z 2 ) 3 is a group of isomorphisms of the Fano plane and has order 168.
Flowcharts
The Fano plane is a small symmetric block diagram , namely, diagram 2- (7,3,1). The points of the diagram are points of the plane, and the blocks of the diagram are lines of the plane. Thus, the Fano plane is an important example of the theory of block diagrams.
Matroid Theory
The Fano plane is one of the important examples in the theory of matroids . The exclusion of the Fano plane as the necessary for the description of some important classes of matroids, such as , and cograph matroids.
If we divide one straight line into three two-point straight lines, we get a “nephane configuration” that can be embedded in the real plane. This is another important example from the theory of matroids, which should be excluded in order to carry out a large number of theorems.
Steiner System
The Fano plane, being a block diagram, is a system of Steiner triples . And in this case, it can be given the structure of a quasigroup . This quasigroup coincides with the multiplicative structure defined by the units of octonions e 1 , e 2 , ..., e 7 (without 1) if the signs of the product of octonions are ignored [3] .
Three-dimensional fun space
The Fano plane can be extended to the three-dimensional case to form the smallest three-dimensional projective space, and it is denoted by PG (3,2). It has 15 points, 35 lines and 15 planes.
- Each plane contains 7 points and 7 lines.
- Each line contains 3 points.
- The planes are isomorphic to the Fano plane.
- Each point belongs to 7 lines.
- Each pair of different points belongs to exactly one straight line.
- Any pair of different planes intersects exactly in one straight line.
See also
- Incidence structure
- Projective geometry
- Transylvanian Lottery
Notes
- ↑ Actually, this is the group PΓL (3,2), but the finite field of order 2 does not have non-identical automorphism, the group turns into PGL (3,2).
- ↑ Manivel, 2006 , p. 457–486.
- ↑ Baez, 2002 , p. 145–205.
Literature
- John Baez. The Octonions. - Bull. Amer. Math. Soc .. - 2002. - T. 39. - DOI : 10.1090 / S0273-0979-01-00934-X . ( Online HTML version )
- JH van Lint, RM Wilson. A Course in Combinatorics. - Cambridge University Press, 1992 .-- S. 197.
- L. Manivel. Configurations of lines and models of Lie algebras // Journal of Algebra. - 2006. - T. 304 , no. 1 . - ISSN 0021-8693 . - DOI : 10.1016 / j.jalgebra.2006.04.029 .
- Burkard Polster (1998) A Geometrical Picture Book , Chapter 1: “Introduction via the Fano Plane”, also pp 21, 23, 27, 29, 71, 73, 77, 112, 115, 116, 132, 174, Springer ISBN 0 -387-98437-2 .
Links
- Weisstein, Eric W. Fano Plane on the Wolfram MathWorld website.
- Finite plane and Fano plane on PlanetMath