Log24

Ron Shaw in "Configurations of planes in PG(5,2)" . . .

"There are some rather weird things happening here."

As G. M. Conwell pointed out in a 1910 paper, the group of all
40,320 permutations of an 8-element set is the same, in an
abstract sense, as the group of all collineations and dualities
of PG(3,2), the projective 3-space over the 2-element field.

This suggests we study the geometry related to the above group's
actions on the 105 partitions of an 8-set into four separate 2-sets.

Note that 105 equals 15×7 and also 35×3.

In such a study, the 15 points of PG(3,2) might correspond (somehow)
to 15 pairwise-disjoint seven-element subsets of the set of 105 partitions,
and the 35 lines of PG(3,2) might correspond (somehow) to 35 pairwise-
disjoint three-element subsets of the set of 105 partitions.

Exercise:  Is this a mere pipe dream?

A search for such a study yields some useful background . . .

.

Taylor's Index of Names  includes neither Conwell nor the
more recent, highly relevant, names Curtis  and Conway .