Fano plane
The Fano plane is a finite projective plane of order 2, the smallest such structure in projective geometry, comprising 7 points and 7 lines where each line contains exactly 3 points, each point lies on exactly 3 lines, any two points determine a unique line, and any two lines intersect at exactly one point.[1][2] Named after the Italian mathematician Gino Fano, who introduced it in 1892 as an example to illustrate the independence of postulates for projective spaces over finite fields, the Fano plane is constructed from the vector space over the finite field GF(2) with two elements, where points correspond to one-dimensional subspaces and lines to two-dimensional subspaces.[3][2] This configuration serves as a foundational example in finite geometry, satisfying the axioms of a projective plane and representing a Steiner triple system S(2,3,7), which encodes combinatorial designs with balanced incomplete block properties.[1] Its automorphism group is the projective special linear group PSL(3,2), a simple group of order 168, highlighting its symmetry and role in group theory.[1] The Fano plane has influenced diverse areas, including coding theory—where it relates to the Hamming code of length 7—and modern combinatorics, such as the Turán number ex(n, Fano plane) in hypergraph extremal theory.[4] Visually, it is often depicted as a triangle with points at vertices, midpoints, and the center, connected by a circumscribed circle as one line, underscoring its non-Euclidean nature over the reals.[1] As the unique projective plane of its order, it exemplifies how finite fields yield discrete geometries distinct from classical continuous ones.[2]Definition and Basic Properties
Geometric Definition
The Fano plane is defined as a projective plane satisfying the following axioms: for any two distinct points, there is exactly one line incident to both; for any two distinct lines, there is exactly one point incident to both; and there exist four points such that no three are collinear.[5] These axioms ensure a structure where lines and points are symmetrically interchangeable, forming the minimal non-degenerate example of such a geometry. As the projective plane of order 2, the Fano plane has exactly n^2 + n + 1 = 7 points and 7 lines, with each line containing n + 1 = 3 points and each point lying on 3 lines.[5] This finite configuration arises uniquely up to isomorphism for order 2, distinguishing it as the smallest instance of a projective plane. The structure is named after the Italian mathematician Gino Fano, who introduced it in 1892 as a model of the projective plane constructed over the field with two elements (the integers modulo 2).[6] A common visual representation of the Fano plane features seven points arranged in a diagram: three points forming an equilateral triangle at the vertices, three additional points at the midpoints of the triangle's sides, and one central point. The seven lines are depicted as the three straight sides of the triangle (each passing through a vertex and the midpoint of the opposite side? No: each side through two vertices and the midpoint), three straight lines connecting a vertex to the midpoint of the opposite side passing through the center, and a curved line (often a circle) passing through the three midpoint points.[5][1]Points and Lines
The Fano plane consists of seven points, conventionally labeled as the set \{1, 2, 3, 4, 5, 6, 7\}. These points are connected by seven lines, where each line is defined as a set of three collinear points forming the following triples: \{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{2,4,6\}, \{2,5,7\}, \{3,4,7\}, and \{3,5,6\}. The incidence relation between points and lines is captured by a $7 \times 7 incidence matrix A, with rows indexed by points $1 to $7 and columns indexed by lines $1 to $7 (ordered as listed above), where A_{i,j} = 1 if point i lies on line j and $0 otherwise. This matrix is: \begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 \end{pmatrix} Each row of the matrix sums to $3, indicating that every point lies on exactly three lines, while each column sums to $3, indicating that every line contains exactly three points; the total number of $1s in the matrix is thus $21, accounting for all incidences in the structure.Incidence Structure
The Fano plane is formalized as an incidence structure in combinatorial design theory, specifically as a balanced incomplete block design (BIBD) with parameters (v, b, r, k, \lambda) = (7, 7, 3, 3, 1), where v=7 is the number of points, b=7 is the number of blocks (lines), r=3 is the number of blocks containing any given point, k=3 is the number of points per block, and \lambda=1 indicates that every pair of distinct points is contained in exactly one block.[1] This configuration realizes a 2-(7,3,1) design, the unique (up to isomorphism) projective plane of order 2, where the incidence relation between points and lines captures the balanced pairwise coverage essential to BIBD properties.[7] The BIBD parameters of the Fano plane satisfy the fundamental consistency equations derived from double counting arguments in design theory: b k = v r, yielding $7 \times 3 = 7 \times 3 = 21, and \lambda (v-1) = r (k-1), yielding $1 \times 6 = 3 \times 2 = 6.[7] These relations confirm the structural integrity of the design, ensuring uniform replication and balance without redundancy or omission in point-block incidences. In this incidence structure, the pairwise balance property holds such that every two distinct points lie on exactly one line, and no three points are collinear except as defined by the blocks themselves, thereby preventing unintended alignments beyond the specified lines.[1] The explicit listings of the seven lines, each comprising three points, exemplify this balance in the concrete realization of the design.[1]Constructions
Homogeneous Coordinates
The Fano plane can be constructed algebraically using homogeneous coordinates over the finite field GF(2, which consists of the elements {0, 1} with arithmetic performed modulo 2, where addition is XOR and multiplication is AND.[8] In this framework, the plane is realized as the projective space PG(2, 2), where points correspond to the one-dimensional subspaces of the three-dimensional vector space (GF(2)^3.[9] Specifically, each point is represented by a non-zero vector in (GF(2)^3, considered up to scalar multiplication by non-zero elements of GF(2; since the multiplicative group of GF(2 is trivial (only 1), there are exactly (2^3 - 1) = 7 distinct points, labeled as [1:0:0], [0:1:0], [0:0:1], [1:1:0], [1:0:1], [0:1:1], and [1:1:1].[1][10] Lines in this coordinate system are the two-dimensional subspaces of (GF(2))^3, each containing (2^2 - 1) = 3 points.[9] Equivalently, a line can be defined as the set of three points whose representative vectors sum to the zero vector in GF(2)^3, reflecting the linear dependence in the subspace.[11] The incidence structure arises naturally: a point lies on a line if its vector is in the subspace spanned by the line. There are 7 such lines, matching the number of points, and each point lies on 3 lines.[1] For example, the line spanned by the basis vectors [1:0:0] and [0:1:0] consists of the points [1:0:0], [0:1:0], and their sum [1:1:0], as these are the non-zero elements of the subspace generated by (1,0,0) and (0,1,0).[10] This construction ensures the projective plane axioms are satisfied, with any two points determining a unique line and any two lines intersecting at a unique point.[9]Group-Theoretic Construction
The Fano plane arises as the projective plane PG(2,2) constructed from the three-dimensional vector space V = (\mathrm{GF}(2))^3 over the finite field with two elements. This space contains $2^3 = 8 vectors, of which the 7 non-zero vectors serve as the points of the plane; each point corresponds to a one-dimensional subspace of V, though in characteristic 2 the non-zero scalar multiples are trivial, identifying each such subspace with its unique non-zero vector.[12][11] The projective special linear group \mathrm{PSL}(3,2) comprises the invertible linear transformations of V with determinant 1, acting on the points via matrix multiplication: for a group element g \in \mathrm{PSL}(3,2) and point represented by non-zero vector v \in V, the action is g \cdot v = g v (modulo scalars, which are trivial here). In characteristic 2, the multiplicative group of \mathrm{GF}(2) is \{1\}, so every invertible matrix has determinant 1, yielding the isomorphism \mathrm{PSL}(3,2) \cong \mathrm{GL}(3,2). This group has order (2^3-1)(2^3-2)(2^3-4) = 7 \cdot 6 \cdot 4 = 168 and acts faithfully and transitively on the 7 points, preserving the projective incidence structure; indeed, it is the full automorphism group of the Fano plane.[12][13] The lines of the Fano plane are defined as the projective lines within PG(2,2), corresponding to the two-dimensional subspaces of V; each such subspace contains $2^2 = 4 vectors, including the zero vector, leaving 3 non-zero vectors that form the points on the line. Three distinct non-zero vectors a, b, c \in V lie on a common line if and only if a + b + c = 0, ensuring linear dependence over \mathrm{GF}(2). The group \mathrm{PSL}(3,2) preserves this incidence because linear transformations map subspaces to subspaces. Equivalently, each line appears as the orbit of a point under the action of a cyclic subgroup of order 3 within \mathrm{PSL}(3,2), such as one generated by a permutation matrix cycling coordinates (e.g., mapping (1,1,0) to (1,0,1) to (0,1,1) and back).[12][11]Projective Plane PG(2,2)
The finite projective plane PG(2, q) of order q, where q is a prime power, is a Desarguesian projective plane constructed over the finite field GF(q). It consists of q² + q + 1 points and the same number of lines, with each line containing q + 1 points and each point incident with q + 1 lines. This structure arises from the projective geometry of dimension 2 over GF(q), ensuring the axioms of a projective plane are satisfied, including the incidence properties that any two distinct points determine a unique line and any two distinct lines intersect at a unique point.[14][15] For q = 2, PG(2, 2) yields the Fano plane, featuring exactly 7 points and 7 lines, each with 3 points. The points of PG(2, 2) are the one-dimensional subspaces (1-flats) of the three-dimensional vector space over GF(2), which has 2³ = 8 vectors total, excluding the zero vector to form (2³ - 1)/(2 - 1) = 7 projective points. Lines correspond to the two-dimensional subspaces (2-flats), each containing (2² - 1)/(2 - 1) = 3 points, and there are likewise 7 such lines. This vector space construction provides a concrete realization, where incidence is defined by subspace containment.[14][11] As a projective plane over the field GF(2), PG(2, 2) is Desarguesian, meaning it satisfies Desargues' theorem: for two triangles in perspective from a point, their corresponding sides intersect at points that are collinear. This property holds inherently in all projective planes derived from fields, distinguishing Desarguesian planes from potential non-Desarguesian ones of higher orders. The Fano plane's adherence to Desargues' theorem underscores its role as the foundational example in finite projective geometry.[16][17] The Fano plane is the unique projective plane of order 2 up to isomorphism, as any such plane must satisfy the defining axioms with exactly 7 points and 7 lines, and all realizations over GF(2) are equivalent. This uniqueness follows from the exhaustive classification of small-order projective planes and the fact that no non-Desarguesian plane exists for order 2.[11][2]Combinatorial Interpretations
Block Design Theory
The Fano plane exemplifies a symmetric balanced incomplete block design (BIBD) within combinatorial design theory, where the points and blocks form a structure satisfying specific incidence relations. In general, a BIBD consists of v points and b blocks such that each block contains k points, each point appears in r blocks, and every pair of distinct points occurs together in exactly \lambda blocks; for the Fano plane, the parameters are v = b = 7, k = r = 3, and \lambda = 1.[18] The symmetry of this design is defined by the equalities b = v and k = r, which ensure that the incidence matrix is symmetric and the structure is self-dual, allowing points and blocks to be interchanged without loss of the design's properties.[18] This self-duality underscores the Fano plane's role as the smallest finite projective plane, embedding it deeply in the theory of symmetric designs. A key theorem in BIBD theory, Fisher's inequality, asserts that b \geq v for any such design, with equality if and only if the design is symmetric. In the Fano plane, b = v = 7 achieves this equality, providing a concrete illustration of the bound and reinforcing its symmetric classification.[19] The proof of Fisher's inequality relies on the rank of the incidence matrix, which equals v and implies the dimension constraint leading to b \geq v.[19] Resolvability in BIBD theory involves partitioning the blocks into parallel classes, each of which covers all points exactly once; a necessary condition for resolvability is that k divides v. For the Fano plane, since 3 does not divide 7, no parallel classes exist, distinguishing it from resolvable designs like affine planes.[20] This lack of resolution highlights a structural limitation unique to projective planes of order 2. Beyond its intrinsic properties, the Fano plane acts as a precursor to higher-dimensional combinatorial constructions, particularly in coding theory and advanced designs. Its incidence structure forms the basis for the extended binary Hamming code of length 8, which extends to the extended binary Golay code of length 24 and ultimately supports the Witt design S(5,8,24), a unique Steiner system with significant applications in symmetry groups and error-correcting codes.[21]Steiner System S(2,3,7)
The Fano plane realizes the unique (up to isomorphism) Steiner triple system of order 7, denoted STS(7). A Steiner triple system of order v, or STS(v), is a collection of 3-element subsets, called triples or blocks, of a v-element point set such that every unordered pair of distinct points is contained in exactly one triple.[22] Such systems exist if and only if v \equiv 1 or $3 \pmod{6}.[22] For v=7, which satisfies this congruence, the Fano plane provides the sole example up to isomorphism, with 7 points and 7 triples.[22][23] Labeling the points as \{1,2,3,4,5,6,7\}, the 7 triples corresponding to the lines of the Fano plane are:- \{1,2,3\}
- \{1,4,5\}
- \{1,6,7\}
- \{2,4,6\}
- \{2,5,7\}
- \{3,4,7\}
- \{3,5,6\}