Edited again think the properties can be devided in the following two groups Special properties of projective geometry of the Fano 3 space mentionable Each point is contained in 7 lines and 7 planes Each line is contained in 3 planes and contains 3 points Each plane contains 7 points and 7 lines Each plane is isomorphic to the Fano plane Every pair of distinct planes intersect in a line A line and a plane intersect in at least one point General properties no need to mention Every pair of distinct points are contained in exactly one line every trio of noncollinear points is in exactly one plane every pair of intersecting lines is in exactly one plane Every pair of distinct planes intersects in exactly one line important is they do intersect it is always in maximum one line I am not sure about Each plane is isomorphic to the Fano plane seems to be a general truth as well and could be removed