Consider . . .
A. The nontrivial analogy between the two parts of the well-known natural
15+15 partition of the 30 labelings of the Fano plane PG(2, 2)
B. The nontrivial analogy between the two parts of the well-known natural
15+15 partition of the 30 planes of the Klein quadric in PG(5, 2)
Are A and B nontrivially analogous? If so, how?
Update of 6:58 PM EDT Oct. 7 . . .
Hint:
Use as labels for PG(2, 2) points the seven nonzero vectors in the
3-space over GF(2), expressed as 001, 010, 011, 100, 101, 110, 111.
Then form three seven-digit vectors by taking the first, second, and third
digit in each 3-digit vector. View these seven-digit vectors as points of
the Klein quadric in PG(5, 2).