(Or, equivalently, as a linear combination of three 2-colorings (into GF(2) as a subfield of GF(4))).
How this works:
Let m be a map into a 4-set.
Represent the elements of the 4-set by the elements
Define f(x,y), where x, y are elements of
f = 1f(a,b) + af(1,b) + bf(1,a)
A modest generalization of the decomposition theorem, and a problem disguised as a query, are given by the 1982 research note below.
View original 1982 note.
The Diamond Theorem
We regard the four-diamond figure D below as a 4x4 array of two-color diagonally-divided square tiles.
Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants.
THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.
For an animated version, click here.
For more on how the decomposition theorem
applies to the diamond theorem, click here.
Orthogonal Latin Squares as Skew Lines.
Page last maintained August 22, 2008; created Nov. 27,