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The Fano plane, a (7 3)-configuration, is unique and is the smallest such configuration. [11] According to a theorem by Steinitz [ 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).
The Fano plane is the projective plane with the fewest points and lines. The smallest 2-dimensional projective geometry (that with the fewest points) is the Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
The Fano plane, discussed below, is denoted by PG(2, 2). The third example above is the projective plane PG(2, 3). The Fano plane. Points are shown as dots; lines are shown as lines or circles. The Fano plane is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines.
The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t = 2 or (recently) t ≥ 2.
The adjunction formula implies that K D = (K X + D)| D = (−(n+1)H + deg(D)H)| D, where H is the class of a hyperplane. The hypersurface D is therefore Fano if and only if deg(D) < n+1. More generally, a smooth complete intersection of hypersurfaces in n-dimensional projective space is Fano if and only if the sum of their degrees is at most n.
The Fano plane cannot be represented in the Euclidean plane using only points and straight line segments (i.e., it is not realizable). This is a consequence of the Sylvester–Gallai theorem, according to which every realizable incidence geometry must include an ordinary line, a line containing only two points. The Fano plane has no such line ...
L 2 (7) ≅ L 3 (2) which acts on the 1 + 2 + 4 = 7 points of the Fano plane (projective plane over F 2); this can also be seen as the action on order 2 biplane, which is the complementary Fano plane. L 2 (11) is subtler, and elaborated below; it acts on the order 3 biplane. [8]
The Fano plane, the projective plane over the field with two elements, is one of the simplest objects in Galois geometry.. Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). [1]