mnaylor@math.fsu.edu     /\       "Abacabadabacaba!"
          /\            /  \            /\          
   /\    /  \    /\    /    \    /\    /  \    /\   
/\/  \/\/    \/\/  \/\/      \/\/  \/\/    \/\/  \/\


"Abacabadabacaba!"  Sounds kind of magical, huh?  Like "Abracadabra!"  
And who knows, maybe that's where the well-known magicians' all-purpose
invocation comes from, because "abacabadabacaba" is certainly a mystical
magical mathematical word.

"abacabadabacaba" is the pattern of mountains in the picture above.
'a' stands for the shortest mountain, 'b' for the next biggest, etc.

To continue the pattern, we'd add a taller mountain ('e') and then repeat
the previous pattern.  If we continue this forever, we'd have a fractal.


                       abacabadabacabaeaabacabada...    /\ 
                         /\                            /  \                            /\   
          /\            /  \            /\            /    \            /\            /  \      
   /\    /  \    /\    /    \    /\    /  \    /\    /      \    /\    /  \    /\    /    \    
/\/  \/\/    \/\/  \/\/      \/\/  \/\/    \/\/  \/\/        \/\/  \/\/    \/\/  \/\/      \/\ . . .



Here's somewhere else you've probably seen this pattern:

|'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'T'|
|   '   |   '   |   '   |   '   |   '   |   '   |   '   |   '   |   '   |   '   |   '   |   '   | |
|       1       2       3       4       5       6       7       8       9      10      11      12 |
!_________________________________________________________________________________________________!

...on any English ruler!  (Metric rulers have more divisions but the 
idea is the same).

Here's where the magic comes in:  we can use "abacabadabacaba" to 
travel through other dimensions without getting lost!

First, lets see how to travel around a square without getting lost.  
Let's call 'a' a movement right or left in the x-direction, and 'b' 
a movement up or down in the y-direction.  If you start in the corner 
of a square, then to visit all the other corners, you could travel 'aba'
(right-down-left, in the picture below) without ever backtracking by mistake.


start ._____A______.
      :            |
      :            |
      :            B
      :            |
      :            |
  end :_____A______!

Got the idea?  Let's move to three dimensions.

Let's call 'c' a movement in the z-direction.  Starting at a corner 
of a cube, you can visit all the other corners with no repeats by 
travelling 'abacaba', as illustrated below.


 ._____A_____,
 |`          :\ 
 | `         : B
 |  `        :  \  
 C   +-----A-----+
 |   :       :   : 
 |__ : _A____:   :
 `   :        \  :
  `  :         B : 
   ` :          \:
    `:_____A_____> 
  start


I think you can see where this goes... if we call 'd' a movement in the
w-direction, we can traverse all the corners of a hypercube with that 
mystical sounding "Abacabadabacaba!".  

Here's the route mapped on a hypercube.  Not all edges are shown... 
what a mess that would be!  Start at either '*', and weave through
the edges.  Be careful, stray off course and you may end up in that
trans-dimensional place where all the missing socks go!


          +_____A_____+      
         /:`          :\     
       D/ : `         : B
       /  :  `        :  \   
      /   :   +-----A-----+  
     +______A_|__,    :   :  
     :`   *___|_A_\___+   :  
     : `  `   C  : B   \  :  
     :  `  `  |  :  \   B :  
     :   +--`-|--A---+   \:  
     :   |   `+____A______+  
     *__ |__A____+   :   ,   
     `   |        \  :  ,    
      `  C         B : ,     
       ` |          \:,      
        `+_____A_____+  


Want to travel a 5th-dimensional hypercube?  Better pack some sandwiches, 
and don't forget your "abacabadabacabaeabacabadabacaba"!
(I'll leave it to you to draw the picture for this one)


--- Michael

Afterthoughts:  I suppose the most powerful magic word of all would be
an "abacaba..." which goes all the way to 'z'... perhaps it can be used
to summon a genie from the 26th dimension.  How many letters would that 
word have?  How long would it take you to say this word?

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