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Original Usenet message from alt.ascii-art, 27 Aug 1997.
Flowsnake (v 1.1)
There was a mistake in the last posting (didya notice?)
It is fixed.

:)





      The Flowsnake
      -------------
asciied by Michael Naylor



Order 1 Flowsnake


________________
\               \
 \               \
  \               \
   \________       \
           /
          / 
         /
________/



Order 2 Flowsnake


            ________
            \       \
  ________   \____   \
  \       \      /   /
   \____   \____/   /   ____
       /            \   \   \
  ____/   ________   \   \   \
 /        \       \   \  /
/   ____   \____   \   \/
\   \   \      /   /
 \   \   \____/   /   ____
  \  /            \  /   /
   \/   ________   \/   /
        \       \      /
         \____   \____/
             /
        ____/

Order 3 Flowsnake
                      ____                    
                 ____ \__ \
                 \__ \__/ / __
                 __/ ____ \ \ \    ____
                / __ \__ \ \/ / __ \__ \
           ____ \ \ \__/ / __ \/ / __/ / __
      ____ \__ \ \/ ____ \/ / __/ / __ \ \ \
      \__ \__/ / __ \__ \__/ / __ \ \ \ \/
      __/ ____ \ \ \__/ ____ \ \ \ \/ / __
     / __ \__ \ \/ ____ \__ \ \/ / __ \/ /
     \ \ \__/ / __ \__ \__/ / __ \ \ \__/
      \/ ____ \/ / __/ ____ \ \ \ \/ ____
         \__ \__/ / __ \__ \ \/ / __ \__ \
         __/ ____ \ \ \__/ / __ \/ / __/ / __
        / __ \__ \ \/ ____ \/ / __/ / __ \/ /
        \/ / __/ / __ \__ \__/ / __ \/ / __/
        __/ / __ \ \ \__/ ____ \ \ \__/ / __
       / __ \ \ \ \/ ____ \__ \ \/ ____ \/ /
       \ \ \ \/ / __ \__ \__/ / __ \__ \__/
        \/ / __ \/ / __/ ____ \ \ \__/
           \ \ \__/ / __ \__ \ \/
            \/      \ \ \__/ / __
                     \/ ____ \/ /
                        \__ \__/
                        __/





                       Flowsnake coloring sheet
                   (makes a great present for Mom!)
                           _
                          / `-_
                         /     `-_
                        /         `-_
                       /             `-_
                      /                 `-_
                     /                     `-_
                    /             ____        `-_              
                   /         ____ \__ \          `-_           
  color this      /          \__ \__/ / __          `-_        
 region a dark   /           __/ ____ \ \ \    ____    `-_      
     color      /           / __ \__ \ \/ / __ \__ \      /      
               /       ____ \ \ \__/ / __ \/ / __/ / __  /   `     
              /   ____ \__ \ \/ ____ \/ / __/ / __ \ \ \/       `  
             /    \__ \__/ / __ \__ \__/ / __ \ \ \ \/             `
            /     __/ ____ \ \ \__/ ____ \ \ \ \/ / __                ` ,
           /     / __ \__ \ \/ ____ \__ \ \/ / __ \/ /                    
          /      \ \ \__/ / __ \__ \__/ / __ \ \ \__/                 ,
         /        \/ ____ \/ / __/ ____ \ \ \ \/ ____
        /            \__ \__/ / __ \__ \ \/ / __ \__ \              ,
       /             __/ ____ \ \ \__/ / __ \/ / __/ / __
      /             / __ \__ \ \/ ____ \/ / __/ / __ \/ /         ,
     /              \/ / __/ / __ \__ \__/ / __ \/ / __/
    /               __/ / __ \ \ \__/ ____ \ \ \__/ / __        ,
   /               / __ \ \ \ \/ ____ \__ \ \/ ____ \/ /
   `-_             \ \ \ \/ / __ \__ \__/ / __ \__ \__/       ,
      `-_           \/ / __ \/ / __/ ____ \ \ \__/
         `-_           \ \ \__/ / __ \__ \ \/               ,   Color this 
            `-_         \/      \ \ \__/ / __                 region a light
               `-_               \/ ____ \/ /             ,     color, or 
                  `-_               \__ \__/                 leave it blank
                     `-_            __/                 ,
                        `-_        / 
                           `-_    /                   ,
                              `-_/
                                 `                  ,
                                    `                
                  cut out around       `          ,
                    the outside           `
                                             `   
                                                `



Flowsnake technical information

The Flowsnake was invented by William Gosper.  It is a space-filling 
fractal curve.  "Space-filling" means that if this process were carried 
out infinitely, this curve would pass through every point in the region 
it occupies.  It is a fractal in that if any "segment" is enlarged, it will 
be identical to the entire curve.  This curve has a fractal dimension of 
log(3)/log(sqrt(7)) = ~1.129.


Detail of construction:

The Flowsnake can be thought of as being composed of seven segments.  
At each iteration, each of these seven segments is replaced with the 
original shape.  (The new segments should have length equal to 1/sqrt(7) 
times the original length; in ascii, of course, we'll make do with an 
approximation).  In the following illustrations, the original vertices 
are labeled to make it easier for you to follow what happens at each stage.




Order 1 Flowsnake


4_______5______6
\               \
 \               \
  \               \
   \3______2       \7
           /
          / 
         /
0______1/



Order 2 Flowsnake


            ________
            \       \
  ________   \____   \6
  \       \      /   /
   \____   \5___/   /   ____
       /            \   \   \
  4___/   ________   \   \   \7
 /        \       \   \  /
/   ____   \____   \2  \/
\   \   \      /   /
 \   \   \3___/   /   ____
  \  /            \  /   /
   \/   ________   \/   /
        \       \      /
         \____   \1___/
             /
        0___/

Order 3 Flowsnake
                      ____                    
                 ____ \__ \
                 \__ \__/ / __
                 __/ ____ \ \ \6   ____
                / __ \__ \ \/ / __ \__ \
           ____ \ \ \__/ / __ \/ / __/ / __
      ____ \__ \ \/ ____ \/ / __/ / __ \ \ \7
      \__ \__/ / __ \__ \__/ / __ \ \ \ \/
      __/ ____ \ \ \5_/ ____ \ \ \ \/ / __
     / __ \__ \ \/ ____ \__ \ \/ / __ \/ /
     \ \ \__/ / __ \__ \__/ / __ \ \ \__/
      \/ ____ \/ / __/ ____ \ \ \2\/ ____
         \__ \__/ / __ \__ \ \/ / __ \__ \
         4_/ ____ \ \ \__/ / __ \/ / __/ / __
        / __ \__ \ \/ ____ \/ / __/ / __ \/ /
        \/ / __/ / __ \__ \__/ / __ \/ / __/
        __/ / __ \ \ \3_/ ____ \ \ \__/ / __
       / __ \ \ \ \/ ____ \__ \ \/ ____ \/ /
       \ \ \ \/ / __ \__ \__/ / __ \__ \__/
        \/ / __ \/ / __/ ____ \ \ \1_/
           \ \ \__/ / __ \__ \ \/
            \/      \ \ \__/ / __
                     \/ ____ \/ /
                        \__ \__/
                       0__/





If you'd like to find out more, try Martin Gardner's "Penrose Tiles to 
Trapdoor Ciphers", W. H. Freeman and Company, 1989.

                                                -- Michael Naylor (8/97)




--
       T         mna...@math.fsu.edu       \`.    T     
 Michae|    T     .--------------.___________) \   |    T
       ! Nay|or   |//////////////|___________[ ]   !  T |
            !     `--------------'           ) (      | !
          http://www.math.fsu.edu/~mnaylor   '-'      !

Original message headers
X-Google-Language: ENGLISH,ASCII-7-bit
X-Google-Thread: f996b,8369afa5afe93013,start
X-Google-Attributes: gidf996b,public
From: Michael Naylor <mna...@math.fsu.edu--remove.to.reply>
Subject: Flowsnake (v 1.1)
Date: 1997/08/27
Message-ID: <340...@math.fsu.edu--remove.to.reply>
X-Deja-AN: 268693125
Organization: Florida State University
Reply-To: mna...@math.fsu.edu--remove.to.reply
Newsgroups: alt.ascii-art
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