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260731-Fano-1892-PG(3,2)…AI_Overview.jpg

260731-Square_Model_of_PG(3,2)-by-Cullinane…AI_Overview.jpg

* See Kirkman's schoolgirl problem in Wikipedia, esp.
the sections "Galois Geometry" and "Spreads and Packing."

The geometry of what might be called "Schoolgirl Space"
is, as the above image "Geometry of the Six Element Set"
indicates, the same as the geometry of the "Hexastigm"
described by Richmond (King's College, Cambridge,
March 30, 1899) —

"On the Figure of Six Points in Space of  Four Dimensions,"
by H. W. Richmond, Quarterly Journal of Pure and Applied
Mathematics, Vol. 31 (1900), pp. 125-160.

This is available on the Web at Scribd.com . . .

https://www.scribd.com/document/424182244/
Richmond-QuarterlyJournal-1900-SixPoints .

Richmond's excellent historical survey, together with the Cullinane
square model of PG(3,2), provide the connection of finite  geometry
to earlier, non-finite geometry that is lacking in some later treatments
of the subject . . . in particular, the connection to finite  geometry of
studies of the classical sixty Pascal lines of the Hexagrammum.