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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.
If P is a finite set, the projective plane is referred to as a finite projective plane. The order of a finite projective plane is n = k – 1, that is, one less than the number of points on a line. All known projective planes have orders that are prime powers. A projective plane of order n is an ((n 2 + n + 1) n + 1) configuration. The smallest ...
In incidence geometry, most authors [16] give a treatment that embraces the Fano plane PG(2, 2) as the smallest finite projective plane. An axiom system that achieves this is as follows: (P1) Any two distinct points lie on a line that is unique. (P2) Any two distinct lines meet at a point that is unique.
Example 1: points and lines of the Euclidean plane (top) Example 2: points and circles (middle), Example 3: finite incidence structure defined by an incidence matrix (bottom) In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects.
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, 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]
A passenger plane burst into flames as it landed at a Tokyo airport Tuesday, after a collision with a coast guard aircraft that killed five people, officials said.. All 379 passengers and crew ...
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.