The symmetry of the line diagrams accounts for the symmetry of the
two-color patterns. (A proof shows that a 2

*n*x2

*n*
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
squares.)

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
S(5,8,24)."

For movable JavaScript versions of these 4x4 patterns, see

The Diamond 16 Puzzle
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
Finite Geometry.