Log24

Saturday, August 6, 2011

Correspondences

Filed under: General,Geometry — Tags: , — m759 @ 2:00 PM

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances "

From "A Four-Color Theorem"

http://www.log24.com/log/pix11B/110806-Four_Color_Correspondence.gif

Figure 1

Note that this illustrates a natural correspondence
between

(A) the seven highly symmetrical four-colorings
      of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
      projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
"points" section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—

http://www.log24.com/log/pix11B/110806-Analysis_of_Structure.gif

Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful.  It yields, as shown, all of the 35 partitions of an 8-element set  (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is  the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.

 

For some applications of the Curtis MOG, see
(for instance) Griess's Twelve Sporadic Groups .

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