May 20, 2009

• The Diamond Connection:

  From Quilt Blocks to the
  Mathieu Group M₂₄

  
  

  Diamonds

  (a traditional
  quilt block):
  
  
  
  _Octads:

  

  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.

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