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crankbuster
Posts:
21
From:
Srilanka
Registered:
7/5/05


Steven Cullinane's "Diamond Theory"
Posted:
Jul 7, 2005 9:54 AM


Reference: http://m759.freeservers.com/  "The Diamond Theorem: Inscribe a white diamond in a black square. Split the resulting figure along its vertical and horizontal midlines into four quadrants so that each quadrant is a square divided by one of its diagonals into a black half and a white half. Call the resulting figure D.
Let G be the group of 24 transformations of D obtained by randomly permuting (without rotating) the four quadrants of D. Let S4 denote the symmetric group acting on four elements. Then
(1) Every Gimage of D has some ordinary or colorinterchange symmetry (see below),
(2) G is an affine group generated by S4 actions on parts of D, and
(3) Results (1) and (2) generalize, through intermediate stages, to symmetry invariance under a group of approximately 1.3 trillion transformations generated by S4 actions on parts of a 4x4x4 cube." 
What does (2) mean? By Cullinane's definition of G, G is isomorphic to S4, the symmetric group on 4 letters, with 24 elements. What does he mean by saying G is generated by S4 actions? Every group generates itself as its own subgroup! This is a theorem?
What does (3) mean? Generalize how? What is "approximately 1.3 trillion"? What "parts" of a 4x4x4 cube? Who is Cullinane trying to fool?
See the "4x4 Case" following the "theorem". Cullinane states that "G is a group of 322,560 permutations". Where does this number come from? Going by his own definition, if 4x4=16 objects are to be permuted (this is the only way to "generalize" his "theorem") then G should be isomorphic to S16, the symmetric group on 16 letters, with 16!=20922789888000 elements. Why 322560? Huh?
Barry for Crank Watch International



