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May 20, 2009 --
This note is to supply some background for the two papers cited in the yellow box below.

The Diamond Connection:
 
From Quilt Blocks to the
Mathieu Group M24
Diamonds

(a traditional
quilt block):

Illustration of a diamond-theorem pattern

Octads:

Octads formed by a 23-cycle in the MOG of R.T. Curtis

Click on illustrations for details.

The connection:

The four-diamond figure is related to the finite geometry PG(3,2). (See "Symmetry Invariance in a Diamond Ring," AMS Notices, February 1979, A193-194.) PG(3,2) is in turn related to the 759 octads of the Steiner system S(5,8,24). (See "Generating the Octad Generator," expository note, 1985.)

The relationship of S(5,8,24) to the finite geometry PG(3,2) has also been discussed in--
  • "A Geometric Construction of the Steiner System S(4,7,23)," by Alphonse Baartmans, Walter Wallis, and Joseph Yucas, Discrete Mathematics 102 (1992) 177-186.
Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."
  • "A Geometric Construction of the Steiner System S(5,8,24)," by R. Mandrell and J. Yucas, Journal of Statistical Planning and Inference 56 (1996), 223-228.
Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."

For the connection of S(5,8,24) with the Mathieu group M24, see the references in The Miracle Octad Generator.
ll>24, see the references in The Miracle Octad Generator.