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The corresponding symmetries on the Fano plane are respectively swapping vertices, rotating the graph, and rotating triangles. Bijection between the Fano plane as field with eight elements minus the origin and the projective line over the field with seven elements. Symmetries are made explicit.
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:
Two triangles and are said to be in perspective centrally if the lines , , and meet in a common point, called the center of perspectivity. They are in perspective axially if the intersection points of the corresponding triangle sides, X = A B ∩ a b {\displaystyle X=AB\cap ab} , Y = A C ∩ a c {\displaystyle Y=AC\cap ac} , and Z = B C ∩ b c ...
There exists a triangle, i.e. three non-collinear points. The lines l and m in the statement of Playfair's axiom are said to be parallel. Every affine plane can be uniquely extended to a projective plane. The order of a finite affine plane is k, the number of points on a line. An affine plane of order n is an ((n 2) n + 1, (n 2 + n) n ...
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 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]
The Fano plane. This particular projective plane is sometimes called the Fano plane. If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called the projective plane of order 2 because it is unique (up to