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In finite geometry, the Fano plane (after Gino Fano) is a
finite projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
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 in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, but they can be given coordinates using the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with two elements. The standard notation for this plane, as a member of a family of
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s, is . Here stands for "
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
", 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 finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study.


Homogeneous coordinates

The Fano plane can be constructed via
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
as the projective plane over the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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 In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
, 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues. The Desargues configuration can be constructed in two dimensions ...
(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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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, 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.


Group-theoretic construction

Alternatively, the 7 points of the plane correspond to the 7 non-identity elements of the group (''Z''2)3 = ''Z''2 × ''Z''2 × ''Z''2. The lines of the plane correspond to the subgroups of order 4, isomorphic to ''Z''2 × ''Z''2. The
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
group GL(3,2) of the group (''Z''2)3 is that of the Fano plane, and has order 168.


Levi graph

As with any incidence structure, the Levi graph of the Fano plane is a
bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V a ...
, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are
incident Incident may refer to: * A property of a graph in graph theory * ''Incident'' (film), a 1948 film noir * Incident (festival), a cultural festival of The National Institute of Technology in Surathkal, Karnataka, India * Incident (Scientology), a ...
. This particular graph is a connected cubic graph (regular of degree 3), has girth 6 and each part contains 7 vertices. It is the Heawood graph, the unique 6-cage.


Collineations

A '' collineation'', ''
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
'', or ''
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
'' of the Fano plane is a permutation of the 7 points that preserves collinearity: that is, it carries
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
points (on the same line) to collinear points. By the Fundamental theorem of projective geometry, the full
collineation group In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is ...
(or
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
, or
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
) is the projective linear group PGL(3,2), also denoted \mathrm_3(\mathbb_2). Since the field has only one nonzero element, this group is isomorphic to the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
PSL(3,2) and the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(3,2). It is also isomorphic to PSL(2,7). This is a well-known group of order 168 = 23·3·7, the next non-abelian simple group after A5 of order 60 (ordered by size). As a
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
acting Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...
on the 7 points of the plane, the collineation group is
doubly transitive A group G acts 2-transitively on a set S if it acts transitively on the set of distinct ordered pairs \. That is, assuming (without a real loss of generality) that G acts on the left of S, for each pair of pairs (x,y),(w,z)\in S\times S with x \neq ...
meaning that any
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
of points can be mapped by at least one collineation to any other ordered pair of points. (See below.) Collineations may also be viewed as the color-preserving automorphisms of the Heawood graph (see figure). \mathbb_8 is a degree three
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of \mathbb_2, so the points of the Fano plane may be identified with \mathbb_8\setminus \. The symmetry group may be written PGL(3,2)=Aut(\mathbb^2\mathbb_2). Similarly, PSL(2,7)=Aut(\mathbb^1\mathbb_7). There is a relation between the underlying objects, \mathbb^2\mathbb_2 and \mathbb^1\mathbb_7 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 of \mathbb_8 will be at the center of the septagon inside. Now label this point as \infty, and pull it backwards to the origin. One can write down a bijection from \mathbb_7\cup \ to \mathbb_8. Set x^\infty=0 and send the slope k \mapsto x^\infty+x^k \in \mathbb_8 \cong \mathbb_2 (x^3+x+1) where now x^k labels the vertices of K_7 with edge coloring, noting that \mathbb_8^* is a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 7. The symmetries of \mathbb^1\mathbb_7 are Möbius transformations, and the basic transformations are reflections (order 2, k \mapsto -1/k), translations (order 7, k \mapsto k+1), and doubling (order 3 since 2^3=1, k \mapsto 2k). The corresponding symmetries on the Fano plane are respectively swapping vertices, rotating the graph, and rotating triangles.


Dualities

A
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the point set and the line set that preserves incidence is called a ''duality'' and a duality of order two is called a ''polarity''. 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. The existence of this polarity shows that the Fano plane is ''self-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.


Cycle structure

The
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
of the 7 points has 6 conjugacy classes. These four cycle structures each define a single 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
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
for a complete list.)
Hence, by the Pólya enumeration theorem, the number of inequivalent colorings of the Fano plane with ''n'' colors is .


Complete quadrangles and Fano subplanes

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 a
configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice bo ...
known as a complete quadrangle. The lines are called ''sides'' and pairs of sides that do not meet at one of the four points are called ''opposite sides''. The points at which opposite sides meet are called ''diagonal points'' and there are three of them. 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 a ''Fano subplane''. A famous result, due to
Andrew M. Gleason Andrew Mattei Gleason (19212008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching at ...
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. Gleason called any projective plane satisfying this condition a ''Fano plane'' thus creating some confusion with modern terminology. To compound the confusion, ''Fano's axiom'' states that the diagonal points of a complete quadrangle are ''never'' collinear, a condition that holds in the Euclidean and real projective planes. Thus, what Gleason called Fano planes do not satisfy Fano's axiom.


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 collineation group, provided each copy can be mapped to any other copy (see
Orbit-Stabiliser theorem In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
). 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 42
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s 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 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 7 ways of selecting a quadrangle of four (unordered) points no three of which are collinear. These four points form the complement of a line, which is the ''diagonal 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 \tbinom = 35 triples of points, seven of which are collinear triples, leaving 28 non-collinear triples or '' triangles''. 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. 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 28 bitangents of a quartic. 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 (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 geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
in which no three consecutive vertices lie on a line, and six symmetries fixing any such hexagon. *There are 84 ways of specifying a
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...
in which no three consecutive vertices lie on a line, and two symmetries fixing any pentagon. The Fano plane is an example of an -configuration, that is, a set of points and 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. According to a theorem by Steinitz configurations of this type can be realized in the Euclidean plane having at most one curved line (all other lines lying on Euclidean lines).


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. With the points labelled 0, 1, 2, ..., 6 the lines (as point sets) are the translates of the (7, 3, 1) planar
difference set In combinatorics, a (v,k,\lambda) difference set is a subset D of size k of a group G of order v such that every nonidentity element of G can be expressed as a product d_1d_2^ of elements of D in exactly \lambda ways. A difference set D is said ...
given by in the group \mathbb / 7\mathbb. With the lines labeled ''ℓ''0, ...,''ℓ''6 the incidence matrix (table) is given by: :


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
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s ''e''1, ''e''2, ..., ''e''7 (omitting 1) if the signs of the octonion products are ignored .


Matroid theory

The Fano matroid F_7 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 of matroids. Excluding the Fano plane as a
matroid minor In the mathematical theory of matroids, a minor of a matroid ''M'' is another matroid ''N'' that is obtained from ''M'' by a sequence of restriction and contraction operations. Matroid minors are closely related to graph minors, and the restrictio ...
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.


PG(3,2)

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 In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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


See also

* Projective configuration *
Transylvania lottery In mathematical combinatorics, the Transylvanian lottery is a lottery where three numbers between 1 and 14, inclusive, are picked by the player for any given ticket, and three numbers are chosen randomly. The player wins if two of their numbers, on ...


Notes


References

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External links

* {{Incidence structures Projective geometry Finite geometry Incidence geometry Configurations (geometry) Matroid theory Dot patterns