Fano Plane
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High Quality Content by WIKIPEDIA articles! In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane with the smallest possible number of points and lines: 7 each. The Fano plane can be constructed via linear algebra as the projective plane over the finite field with two elements. One can similarly define 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…mehr

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High Quality Content by WIKIPEDIA articles! In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane with the smallest possible number of points and lines: 7 each. The Fano plane can be constructed via linear algebra as the projective plane over the finite field with two elements. One can similarly define 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. This can be done in such a way that for every two points p and q, the third point on line pq has the label formed by adding the labels of p and q modulo 2. In other words, the points of the Fano plane correspond to the non-zero points of the finite vector space of dimension 3 over the finite field of order 2.