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Topic: Steven Cullinane's "Diamond Theory"
Replies: 6   Last Post: Jul 12, 2005 3:58 AM

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Posts: 21
From: Srilanka
Registered: 7/5/05
Steven Cullinane's "Diamond Theory"
Posted: Jul 7, 2005 9:54 AM
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"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 G-image of D has some ordinary or color-interchange 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

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?

Crank Watch International

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