May 28, 2007

• Notes on the I Ching:

  Space-Time

  and a Finite Model
  
  Notes by Steven H. Cullinane
  May 28, 2007
  
  Part I: A Model of Space-Time
  

  The following paper includes a figure illustrating
  Penrose’s model of  “complexified, compactified Minkowski
  space-time as the Klein quadric in complex projective 5-space.”

   

  Click on picture to enlarge.

  For some background on the Klein quadric and space-time, see Roger Penrose, “On the Origins of Twistor Theory,” from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.

  
  Part II: A Corresponding Finite Model

  The Klein quadric also occurs in a finite model of projective 5-space.  See a 1910 paper:

  G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.

  Conwell discusses the quadric, and the related Klein correspondence, in detail.  This is noted in a more recent paper by Philippe Cara:
  

  As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight heptads.  These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M₂₄.

  
  Related material:

  The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.

  The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.

  There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a
  4x4x4 cube.  This correspondence leads to a natural way to generate the affine group AGL(6,2).  This may in turn be viewed as a group of over a trillion natural
  transformations of the 64 hexagrams.

  See

  Finite Geometry of
  the Square and Cube

  

  and

  Geometry of the I Ching.

  “Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching
  into the Glass Bead Game.  Elder Brother laughed.  ‘Go ahead
  and try,’ he exclaimed.  ‘You’ll see how it turns out. 
  Anyone can create a pretty little bamboo garden in the world.  But
  I doubt that the gardener would succeed in incorporating the world in
  his bamboo grove.’”

  – Hermann Hesse, The Glass Bead Game,

    translated by Richard and Clara Winston

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