Fano plane

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. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is PG(2, 2). Here, PG stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one).

The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study.

In a separate usage, a Fano plane is a projective plane that never satisfies Fano's axiom; in other words, the diagonal points of a complete quadrangle are always collinear. "The" Fano plane of 7 points and lines is "a" Fano plane.

A standard visualization of the Fano plane draws its seven points as the vertices, edge midpoints, and centroid of an equilateral triangle, and its seven lines as the three sides and three symmetry axes of the triangle, together with a circle through the three edge midpoints. However, the visual differences between the positions of these points and the shapes of these lines are merely an artifact of the visualization. As an abstract structure, the Fano plane is highly symmetric, with symmetries that take any point to any other point or that take any line to any other line.