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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 projective plane of order 2 is called the Fano plane. See also the article on finite geometry. Using the vector space construction with finite fields there exists a projective plane of order N = p n, for each prime power p n. In fact, for all known finite projective planes, the order N is a prime power. [citation needed]
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. 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
A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem. The Fano plane is the projective plane with the fewest points ...
In finite geometry, PG(3, 2) is the smallest three-dimensional projective space.It can be thought of as an extension of the Fano plane.It has 15 points, 35 lines, and 15 planes. [1]
In algebraic geometry, a Fano variety, introduced by Gino Fano (Fano 1934, 1942), is an algebraic variety that generalizes certain aspects of complete intersections of algebraic hypersurfaces whose sum of degrees is at most the total dimension of the ambient projective space.
The Levi graph of the Fano plane is the Heawood graph, in which the triangles of the Fano plane are represented by 6-cycles. The 28 6-cycles of the Heawood graph in turn correspond to the 28 vertices of the Coxeter graph. [6]