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In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point.
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
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.
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 ...
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 number n is called the order of the affine plane. All known finite affine planes have orders that are prime or prime power integers. The smallest affine plane (of order 2) is obtained by removing a line and the three points on that line from the Fano plane.
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, 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]