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SLICING_THE_CUBE

                                                   February 18, 2010

  Thinking about

      3D_TIC

  An observation I thought was neat
  is that the mapping of the cube as
  those particular three boards is
  arbitrary, and you could slice the
  cube up along another direction and
  have a precisely equivalent layout.

These two sets of three boards are equivalent:

     I                II             III

    o |   | x       o |   | o       o |   | x    C
   ---+---+---     ---+---+---     ---+---+---
      | x |         o | x | o         |   |      B
   ---+---+---     ---+---+---     ---+---+---
    x |   |           | x | o         |   | x    A


     A               B               C

    x |   |           | x |         o |   | x     I
   ---+---+---     ---+---+---     ---+---+---
      | x | o       o | x | o       o |   | o     II
   ---+---+---     ---+---+---     ---+---+---
      |   | x         |   |         o |   | x     III



What's interesting here is you can cover
three dimensions by just thinking about
two at a time... and you can keep              (Even with the diagonals
switching which two you consider               through the center, you
primary.  This is a tool for grasping a        can think about planes in
3-D situation without holding it all in        those directions as well.)
your head at once.


And this is a basic skill that gets more
useful with higher dimensions:

For 4-D, you just consider one
set of 3 dimensions at a time.

So you can play 4-D tic-tac-toe as well...
draw three sets of three boards to map the
hyper-cube, and think in terms of one stack
of three boards at a time.

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