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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.
A projective plane is a linear space in which: Every pair of distinct lines meet in exactly one point, and that satisfies the non-degeneracy condition: There exist four points, no three of which are collinear. There is a bijection between P and L in a projective plane. If P is a finite set, the projective plane is referred to as a finite ...
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.
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.
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.
Fano Plane (7 points and 7 lines) Fano went on to describe finite projective spaces of arbitrary dimension and prime orders. In 1907 Gino Fano contributed two articles to Part III of Klein's encyclopedia. The first (SS. 221–88) was a comparison of analytic geometry and synthetic geometry through their historic development in the 19th century ...
The Airbus A350 was landing at Tokyo’s Haneda airport when it was in collision with a much smaller plane working for the Japanese coastguard in earthquake relief. Tragically, five of the six ...
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.