Infinite geometry, theFano plane (named afterGino Fano) is afinite 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 inEuclidean geometry, but they can be given coordinates using thefinite field with two elements. The standard notation for this plane, as a member of a family ofprojective spaces, isPG(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).

The Fano plane is an example of a finiteincidence structure, so many of its properties can be established usingcombinatorial techniques and other tools used in the study ofincidence 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 satisfiesFano's axiom; in other words, the diagonal points of acomplete quadrangle are always collinear.^[1] "The" Fano plane of 7 points and lines is "a" Fano plane.

The Fano plane can be constructed vialinear algebra as theprojective plane over thefinite 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 viahomogeneous 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 pointsp andq, the third point on linepq has the label formed by adding the labels ofp andq 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 finitevector space of dimension 3 over the finite field of order 2.

Due to this construction, the Fano plane is considered to be aDesarguesian plane, even though the plane is too small to contain a non-degenerateDesargues 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 theinner 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 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.

Alternatively, the 7 points of the plane correspond to the 7 non-identity elements of thegroup(Z₂)³ = Z₂ × Z₂ × Z₂. The lines of the plane correspond to the subgroups of order 4, isomorphic toZ₂ × Z₂. Theautomorphism groupGL(3, 2) of the group (Z₂)³ is that of the Fano plane, and has order 168.

As with any incidence structure, theLevi graph of the Fano plane is abipartite graph, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line areincident. This particular graph is a connectedcubic graph (regular of degree 3), hasgirth 6 and each part contains 7 vertices. It is theHeawood graph, the unique6-cage.^[2]

Acollineation,automorphism, orsymmetry of the Fano plane is a permutation of the 7 points that preserves collinearity: that is, it carriescollinear points (on the same line) to collinear points. By theFundamental theorem of projective geometry, the fullcollineation group (orautomorphism group, orsymmetry group) is theprojective linear groupPGL(3, 2),^[a]Hirschfeld 1979, p. 131^[3]

This is awell-known group of order 168 = 2³·3·7, the next non-abelian simple group afterA₅ of order 60 (ordered by size).

As apermutation groupacting on the 7 points of the plane, the collineation group isdoubly transitive meaning that anyordered pair of points can be mapped by at least one collineation to any other ordered pair of points.^[4] (See below.)

Collineations may also be viewed as the color-preserving automorphisms of theHeawood graph (see figure).

F₈ is a degree-threefield extension ofF₂, so the points of the Fano plane may be identified withF₈ ∖ {0}. The symmetry group may be writtenPGL(3, 2) = Aut(P²F₂). Similarly,PSL(2, 7) = Aut(P¹F₇). There is a relation between the underlying objects,P²F₂ andP¹F₇ called the Cat's Cradle map. Color the seven lines of the Fano plane ROYGBIV, place your fingers into the two dimensional projective space in ambient 3-space, and stretch your fingers out like the children's game Cat's Cradle. You will obtain a complete graph on seven vertices with seven colored triangles (projective lines). The missing origin ofF₈ will be at the center of the septagon inside. Now label this point as ∞, and pull it backwards to the origin. One can write down a bijection fromF₇ ∪ {∞} toF₈. Setx^∞ = 0 and send the slopek ↦x^∞ +x^k ∈F₈ ≅F₂[x] / (x³ +x + 1), where nowx^k labels the vertices ofK₇ withedge coloring, noting thatF^×
₈ is acyclic group of order 7. The symmetries ofP¹F₇ areMöbius transformations, and the basic transformations are reflections (order 2,k ↦ −1/k), translations (order 7,k ↦k + 1), and doubling (order 3 since2³ = 1,k ↦ 2k). The corresponding symmetries on the Fano plane are respectively swapping vertices, rotating the graph, and rotating triangles.

Abijection between the point set and the line set that preserves incidence is called aduality and a duality of order two is called apolarity.^[5]

Dualities can be viewed in the context of the Heawood graph as color reversing automorphisms. An example of a polarity is given by reflection through a vertical line that bisects the Heawood graph representation given on the right.^[6] The existence of this polarity shows that the Fano plane isself-dual. This is also an immediate consequence of the symmetry between points and lines in the definition of the incidence relation in terms of homogeneous coordinates, as detailed in an earlier section.

Thepermutation group of the 7 points has 6conjugacy classes.

These fourcycle structures each define a single conjugacy class:

• The identity permutation
• 21 permutations with two2-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 toB,B toC,C toD. ThenD is on the same line asA andB.
• A maps toB,B toC,C toD. ThenD is on the same line asA andC.

(Seehere for a complete list.)

The number of inequivalent colorings of the Fano plane with${\displaystyle n}$ colors can be calculated by plugging the numbers of cycle structures into thePólya enumeration theorem. This number of colorings is$\frac{n^7+21n^5+98n^3+48n}{168}$(sequenceA241929 in theOEIS).

In any projective plane a set of four points, no three of which are collinear, and the six lines joining pairs of these points is aconfiguration known as acomplete quadrangle. The lines are calledsides and pairs of sides that do not meet at one of the four points are calledopposite sides. The points at which opposite sides meet are calleddiagonal points and there are three of them.^[7]

If this configuration lies in a projective plane and the three diagonal points are collinear, then the seven points and seven lines of the expanded configuration form a subplane of the projective plane that is isomorphic to the Fano plane and is called aFano subplane.

A famous result, due toAndrew M. Gleason states that if every complete quadrangle in a finite projective plane extends to a Fano subplane (that is, has collinear diagonal points) then the plane is Desarguesian.^[8] Gleason called any projective plane satisfying this condition aFano plane thus creating some confusion with modern terminology. To compound the confusion,Fano's axiom states that the diagonal points of a complete quadrangle arenever collinear, a condition that holds in the Euclidean and real projective planes. Thus, what Gleason called Fano planes do not satisfy Fano's axiom.^[9]

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 (seeOrbit-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.

• 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 42ordered pairs 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 21unordered 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 21flags 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 7 ways of selecting aquadrangle of four (unordered) points no three of which are collinear. These four points form the complement of a line, which is thediagonal 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.
• There are${\displaystyle (\genfrac{}{}{0ex}{}{7}{3})}$ = 35 triples of points, seven of which are collinear triples, leaving 28 non-collinear triples ortriangles. 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 28bitangents 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.
• There are 28 ways of selecting a point and a line that are not incident to each other (ananti-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 top and that do not belong tol 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 ahexagon in which no three consecutive vertices lie on a line, and six symmetries fixing any such hexagon.
• There are 84 ways of specifying apentagon in which no three consecutive vertices lie on a line, and two symmetries fixing any pentagon.

The Fano plane is an example of an(n₃)-configuration, that is, a set ofn points andn 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 bySteinitz^[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]

The Fano plane is a smallsymmetric block design, specifically a2-(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.^[14] As such it is a valuable example in (block) design theory.

With the points labelled 0, 1, 2, ..., 6 the lines (as point sets) are the translates of the(7, 3, 1) planardifference set given by{0, 1, 3} in the groupZ / 7Z.^[14] With the lines labeledℓ₀, ...,ℓ₆ theincidence matrix (table) is given by:

Point Line	0	1	2	3	4	5	6
ℓ0	1	1	0	1	0	0	0
ℓ1	0	1	1	0	1	0	0
ℓ2	0	0	1	1	0	1	0
ℓ3	0	0	0	1	1	0	1
ℓ4	1	0	0	0	1	1	0
ℓ5	0	1	0	0	0	1	1
ℓ6	1	0	1	0	0	0	1

The Fano plane, as a block design, is aSteiner triple system.^[15] As such, it can be given the structure of aquasigroup. This quasigroup coincides with the multiplicative structure defined by the unitoctonionse₁,e₂, ...,e₇ (omitting 1) if the signs of the octonion products are ignored (Baez 2002).

The Fano matroidF₇ is formed by taking the Fano plane's points as the ground set, and the three-element noncollinear subsets as bases.

The Fano plane is one of the important examples in the structure theory ofmatroids. Excluding the Fano plane as amatroid minor is necessary to characterize several important classes of matroids, such asregular,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.

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

• 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 isisomorphic 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.

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