November 24, 2006

• "Simplify, simplify..."

  Galois's Window:

  Geometry
  from Point
  to Hyperspace
  
  by Steven H. Cullinane

   
  Euclid is "the most famous
  geometer ever known
  and for good reason:
   
  for millennia it has been
  his window
   
  that people first look through
  when they view geometry."

   
  -- Euclid's Window:
  The Story of Geometry
  from Parallel Lines
  to Hyperspace,

  by Leonard Mlodinow
  

  "...the source of
  all great mathematics
  is
  the special case,
  the concrete example.
  It is frequent in mathematics
  that every instance of a
    concept of
  seemingly
  great generality is
  in essence the same as
  a small and concrete
  special case."

  -- Paul Halmos in
  I Want To Be a Mathematician

  Euclid's geometry deals with affine
  spaces of 1, 2, and 3 dimensions
  definable over the field
  of real numbers.

  Each of these spaces
  has infinitely many points.

  Some simpler spaces are those
  defined over a finite field--
  i.e., a "Galois" field--
  for
  instance, the field
  which has only two
  elements, 0 and 1, with
  addition and multiplication
  as follows:

  + 0 1 0 0 1 1 1 0

  * 0 1 0 0 0 1 0 1

  We may picture the smallest
  affine spaces over this simplest
  field by using square or cubic
  cells as "points":

  

  From these five finite spaces,
  we may, in accordance with
  Halmos's advice,

  select as "a small and
  concrete special case"
  the 4-point affine plane,
  which we may call

  

  Galois's Window.

  The interior lines of the picture

  are by no means irrelevant to
  the space's structure, as may be
  seen by examining the cases of
  the above Galois affine 3-space
  and Galois affine hyperplane

  in greater detail.

  For more on these cases, see

  The Eightfold Cube,

  Finite Relativity,
  
  The Smallest Projective Space,

  Latin-Square Geometry, and

  Geometry of the 4x4 Square.

  (These documents assume that
  the reader is familar with the
  distinction between affine and
  projective geometry.)

  These 8- and 16-point spaces
  may be used to
  illustrate the action of Klein's
  simple group of order 168
  and the action of
  a subgroup of 322,560 elements

  within the large Mathieu group.

  The view from Galois's window

  also includes aspects of

  quantum information theory.

  For links to some papers
  in this area, see
    Elements of Finite Geometry.

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