For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of …
Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).
This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .
Update, morning of the following day (7:07 ET) — related material:
See also “56 spreads” in this journal.