The Fano Plane — PG(2,2)

The unique projective plane of order 2: 7 points, 7 lines, 3 points per line, 3 lines per point. Every pair of points determines a unique line; every pair of lines meets at a unique point.

Lines (click to highlight)

Symmetry group: GL(3,2) = PSL(2,7), order 168

Octonion connection: Each line {i,j,k} gives an octonion triple e_i·e_j = e_k (Fano mnemonic)

Error correction: Fano plane is the [7,4,3] Hamming code — dual of [7,3,4] simplex code

Collinearity: Any 3 of 7 points are either collinear (on one of 7 lines) or form a "quadrangle"

Characteristic: Only finite projective plane with all lines of minimum size (order 2)