Fano plane

The Fano plane

Duality in the Fano plane: Each point corresponds to a line and vice versa.

In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point. The standard notation for this plane, as a member of a family of projective spaces, is $PG(2,2)$ where $PG$ stands for "Projective Geometry", the first parameter is the geometric dimension and the second parameter is the order.

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. 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 codes 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 codes 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, 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 code has exactly two nonzero bits.

Symmetries

A collineation of the Fano plane corresponding to the 3-bit Gray code permutation

A permutation of the seven points of the Fano plane 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 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 general linear group GL(3,2) (which is equal to PGL(3,2) because the field has only one nonzero element). It consists of 168 different permutations.

The automorphism group is made up of 6 conjugacy classes.
All cycle structures except the 7-cycle uniquely define a conjugacy class:

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

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

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

See Fano plane collineations for a complete list.

Hence, by the Pólya enumeration theorem, the number of inequivalent colorings of the Fano plane with n colors is:

{1 \over 168}\left(n^7 + 21 n^5 + 98 n^3 + 48 n\right).

Configurations

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 symmetry group.

There are 7 points, and 24 symmetries fixing any point.
There are 7 lines, and 24 symmetries fixing any line.
There are 7 ways of selecting a quadrangle of four (unordered) points no three of which are collinear, and 24 symmetries that fix any such quadrangle. These four points form the complement of a line, which is the diagonal line of the quadrangle.
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.
There are 21 flags 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.
There are 28 triangles, which correspond one-for-one with the 28 bitangents of a quartic (Manivel 2006). For each triangle there are six symmetries fixing it, one for each permutation of the points within the triangle.
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 hexagon in which no three consecutive vertices lie on a line, and six symmetries fixing any such hexagon.
There are 42 ordered pairs of points, and again each may be mapped by a symmetry onto any other ordered pair. For any ordered pair there are 4 symmetries fixing it.
There are 42 ways of selecting a quadrangle of four cyclically ordered points no three of which are collinear, and four symmetries that fix any such ordered quadrangle. For each not-oriented quadrangle there are two cyclic orders.
There are 84 ways of specifying a triangle together with one point on that triangle, each of which has two symmetries fixing it.
There are 84 ways of specifying a pentagon in which no three consecutive vertices lie on a line, and two symmetries fixing any pentagon.
There are 168 different ways of specifying a triangle together with an ordering for its three points, and only the identity symmetry fixes this configuration.

Group-theoretic construction

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.

Block design theory

The Fano plane is a small symmetric block design, specifically a 2-(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 is a valuable example in (block) design theory.

Matroid theory

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 matroid minor is necessary to characterize several important classes of matroids, such as regular, graphic, and cographic ones.

If you break one line apart into three 2-point lines you obtain the "non-Fano configuration", which can be embedded in the real plane. It is another important example in matroid theory, as it must be excluded for many theorems to hold.

Steiner system

Main article: Steiner system

The Fano plane, as a block design, is a Steiner triple system. As such, it can be given the structure of a quasigroup. This quasigroup coincides with the multiplicative structure defined by the unit octonions e₁, e₂, ..., e₇ (omitting 1) if the signs of the octonion products are ignored (Baez 2002).

Fano three-space

PG(3,2) but not all the lines are drawn

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. It also has the following properties:

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 isomorphic to the Fano plane
Every pair of distinct planes intersect in a line
A line and a plane not containing the line intersect in exactly one point

Notes

↑ Actually it is PΓL(3,2), but since the finite field of order 2 has no non-identity automorphisms, this becomes PGL(3,2).

References

Baez, John (2002), "The Octonions", Bull. Amer. Math. Soc., 39 (2): 145–205, doi:10.1090/S0273-0979-01-00934-X (Online HTML version)
van Lint, J. H.; Wilson, R. M. (1992), A Course in Combinatorics, Cambridge University Press, p. 197
Manivel, L. (2006), "Configurations of lines and models of Lie algebras", Journal of Algebra, 304 (1): 457–486, doi:10.1016/j.jalgebra.2006.04.029, ISSN 0021-8693
Weisstein, Eric W. "Fano Plane". MathWorld.

"Finite plane and Fano plane". PlanetMath.
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 .

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