Log24

Wednesday, September 5, 2018

Multifaceted Narrative

Filed under: General,Geometry — m759 @ 8:19 PM

http://www.log24.com/log/pix18/180905-To_build_the_narrative-Galerie_St_Etienne.gif

http://www.log24.com/log/pix18/180905-Messier-Objects.gif

See also, in this  journal, 23-cycle.

Update of Sept. 6, 2018, 9:05 AM ET:  "The Cubist Method"

Multifaceted narrative by James Joyce —

http://www.log24.com/log/pix18/180819-Joyce-Possible_Permutations-Cambridge_Companion-2004-p168.gif

Multifaceted structures in pure mathematics, from Plato and R. T. Curtis  —

Counting symmetries with the orbit-stabilizer theorem

Sunday, September 3, 2017

Broomsday Revisited

Filed under: General,Geometry — m759 @ 9:29 AM

Ivars Peterson in 2000 on a sort of conceptual art —

" Brill has tried out a variety of grid-scrambling transformations
to see what happens. Aesthetic sensibilities govern which
transformation to use, what size the rectangular grid should be,
and which iteration to look at, he says. 'Once a fruitful
transformation, rectangle size, and iteration number have been
found, the artist is in a position to create compelling imagery.' "

"Scrambled Grids," August 28, 2000

Or not.

If aesthetic sensibilities lead to a 23-cycle on a 4×6 grid, the results
may not be pretty —

From "Geometry of the 4×4 Square."

See a Log24 post, Noncontinuous Groups, on Broomsday 2009.

Wednesday, May 20, 2009

Wednesday May 20, 2009

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM
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.

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