Finite
Geometry
Notes

Finite Relativity: The Triangular Version
(Continued from 1986)
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them. —
H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered ntuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such ntuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: A 4×4 array. The invariant structure: The following set of 15 partitions of the frame into two 8sets.
A representative coordinatization:
0000
0001 0010 0011
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4tuples over GF(2). 
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them. —
H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered ntuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such ntuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle. The invariant structure: The following set of 15 partitions of the frame into two 8sets.
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4tuples over GF(2). 
For
some background on the triangular version,
see the SquareTriangle Theorem,
noting particularly the linkedto coordinatization picture.