The symmetry of the line diagrams accounts for the symmetry of the
two-color patterns. (A proof shows that a 2n
two-color triangular half-squares pattern with such line diagrams must
have a 2x2 center with a symmetry, and that this symmetry must be
shared by the entire pattern.)
Among the 35 structures of the 840 4x4 arrays of tiles, orthogonality
(in the sense of Latin-square orthogonality) corresponds to skewness of
lines in the finite projective space PG(3,2). This was stated by the
author in a 1978 note. (The note apparently had little effect. A
quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A.
Thas wrote that skew (i.e., nonintersecting) lines in a projective
space seem "at first sight not at all related" to orthogonal Latin
We can define sums and products so that the G-images of D generate an
ideal (1024 patterns characterized by all horizontal or vertical "cuts"
being uninterrupted) of a ring of 4096 symmetric patterns. There is an
infinite family of such "diamond" rings, isomorphic to rings of
matrices over GF(4).
The proof uses a decomposition technique for functions into a finite
field that might be of more general use.
The underlying geometry of the 4x4 patterns is closely related to the
Miracle Octad Generator of R. T. Curtis-- used in the construction of
the Steiner system S(5,8,24)-- and hence is also related to the Leech
lattice, which, as Walter Feit has remarked, "is a blown up version of
and the easier Kaleidoscope Puzzle
The above is an expanded version of Abstract 79T-A37, "Symmetry
invariance in a diamond ring," by Steven H. Cullinane, Notices of
the American Mathematical Society
, February 1979, pages A-193, 194.
For a discussion of other cases of the theorem, click here
For some context, see Reflection Groups in