Fano plane

Description

The Fano plane has the $7$ points $1, 2, \dots, 7$ and the $7$ lines $$\set{1,2,3},\ \set{1,4,5},\ \set{1,6,7},\ \set{2,4,6},\ \set{2,5,7},\ \set{3,4,7},\ \set{3,5,6}.$$ Check: each line has $3$ points, so the lines cover $7 \cdot 3 = 21$ pairs of points. These $21$ pairs are distinct, and there are $\binom{7}{2} = 21$ pairs in all, so every two points lie on exactly one line. Every two lines meet in exactly one point, as one sees from the list. So the Fano plane is a finite projective plane of order $2$ and a Steiner system $S(2, 3, 7)$.