3/8                    5/16             1/4           3/16
  '-,                    \               :    7/32    :-                
    '-                    ~      9/32    '     :     .~               .,1/8
      '`,                  ~,      -,     `    :    .~     5/32     _-' 
         -,                 ",      ~,    ', , ,   :~    .-      ..-`   
           -,      11/32     'e      ~,  . ",a,\.`.`.  .,`     _-'      
             -,       ^,      '\   .  '.  `-$"@@LL)v'.,'     . `        
               -,      '\       ~,  ~, ,`- -,,%@@F\(-' ,- .,       3/32
                '-,      ~,  -,  '. .'e'\\;"@F*`*@@/'_-` r' ,  _.-^ 
7/16    13/32     '.   _  ",  '. - `:_/(,e_b@L, ./@mmL'~cm)e.-'`        
  -,,       -,      ~, '- ," ', -,-)r@@*Y*"^^`   '^^"*b/Y@@`"--  ,_,--''1/16
    '--.      -.  `. .ra:c`c`.' ~//md@F^              '-:@^_^c-^'       
         .,  .  -_ .`-'@@)ed@dbL`\\@*`                   "*@/--``     _,1/32
15/32_    '-  -,-,-_:-.(@@"**^**@$@F                     'Y@^%^^^'^^'^  
      `-.'--`d,7/d'/dm/@F^       ^@                      .@"~----       
1/2----":bLbbFb@@bmd@*^^`         '  RF(2){FS(2)}x3Sa  -a/"*-'---------- 0
       d;'.`"*^""-^F$@mme        ._                      "mTrL___       
17/32-^` ' .~ _~-- ;^^;@@bc__.___d@,                      @@,m__        
        . -  -`  ,` -`(a@@@@@d@@^`$b_                    .T@Lm_ ^^^`^^--31/32
      .-'      ,` .,'-(@Y"(-"\```-:@@a,                .a@@*C-,^`       
  ,--`      _'`   '  -``/ `.-., `-\T($@`.,,        .__a__@bF__,`^---,_  
9/16       -`      /'  -  .'   r .-'d@@@*YbFmm- -vmdbF"^@@@/"_'       `^15/16
      19/32      .-      _~  .~   ,'-`/~)``_@b,,_a@m``;-`. -. `--.      
               _-`      .`      _-  ." ~.,`~TY@b@FF``-'-, `.      `     
             _-`      ./       .`  -`  r' -.\d@@@@c^  . '   ` ,   29/32 
           _/     21/32       /      ."  -'.'(F"\ - `  `.     '-,       
         _-                 .~      :~    .~ ' `   ~,    -,      ` ,    
       _-`                 .~      :-     -    .    ~,  27/32      ' ,  
     --`                  a~    23/32    .`    '     -,               7/8
   .-                    .`              .   25/32    ~,                
5/8                    11/16            3/4           13/16
A one-to-one mapping of the unit circle onto the boundary of the Mandelbrot set yields a simple method of naming any point on the boundary. The external angle is a real number between 0 and 1. There are an infinite number of possible continuous mappings of (0,1) onto the boundary of the Mandelbrot Set, but the one which is most useful is based on the orbit dynamics (search Google for 'mu-ency iteration orbit dynamics')

If the external angle is expressed as a binary number, useful patterns can be seen. These are easy to see in an external angle plot.

If a rational external angle is expressed as a fraction A/B, the values of the numerator A and denominator B yield useful information. Of particular note, if the denominator is 2N-1, the external angle leads to the root of a mu-atom of period N. All other denominators correspond to external angles that lead to branch points, filament tips, etc.

An external angle can be used as a name for the corresponding point on the Mandelbrot Set's boundary. This has an advantage over more complex naming systems in that a computer can automatically find a feature given its name. However, such names cannot be used easily by humans without the aid of external-angle software.

Of course, external angles are often used as part of a more elaborate naming system; in the R2 system they are used in the filament subset operator.

The illustration above (based on the image of R2F(1/2B1)FS(2)FS(2)FS(2)S) shows the "first" 32 external angles.

Here is a table giving the corresponding FS operators:

External Angle   FS suffix                     Abbreviated FS suffix
 0/1             FS(0)                         FS(0)
 1/32            FS((1/6B1)t)                  FS(6)
 1/16            FS((1/5B1)t)                  FS(5)
 3/32            FS((1/4B2)t)                  FS(4B2)
 1/8             FS((1/4B1)t)                  FS(4)
 5/32            FS((1/3(1/3B1)B1)t)           FS(3(3))
 3/16            FS((1/3B2)t)                  FS(3B2)
 7/32            FS((1/3B1)FS(2)FS(0)t)        FS((3)F(2)(0))
 1/4             FS((1/3B1)t)                  FS(3)
 9/32            FS((1/3(2/3B1)B1)t)           FS(3(2/3))
 5/16            FS((2/5B1)t)                  FS(2/5)
11/32            FS((1/2(1/4B1)B1)t)           FS(2(4))
 3/8             FS((1/2(1/3B1)B1)t)           FS(2(3))
13/32            FS((1/2(1/2(1/3B1)B1)B1)t)    FS(2(2(3)))
 7/16            FS((1/2B1)SF(1/3B1)t)         FS(2SF(3))
15/32            FS((1/2B1)SF(2)SF((1/3B1)t)t) FS((2)F(2)S.F(3))
 1/2             FS((1/2B1)t)                  FS(2)
17/32            FS((1/2B1)SF(2)SF((2/3B1)t)t) FS((2)F(2)S.F(2/3))
 9/16            FS((1/2B1)SF(2/3B1)t)         FS(2SF(2/3))
19/32            FS((1/2(1/2(2/3B1)B1)B1)t)    FS(2(2(2/3)))
 5/8             FS((1/2(2/3B1)B1)t)           FS(2(2/3))
21/32            FS((1/2(3/4B1)B1)t)           FS(2(3/4))
11/16            FS((3/5B1)t)                  FS(3/5)
23/32            FS((2/3(1/3B1)B1)t)           FS(2/3(3))
 3/4             FS((2/3B1)t)                  FS(2/3)
25/32            FS((2/3B1)FS(2)FS(0)t)        FS((2/3)F(2)(0))
13/16            FS((2/3B2)t)                  FS(2/3B2)
27/32            FS((2/3(2/3B1)B1)t)           FS(2/3(2/3))
 7/8             FS((3/4B1)t)                  FS(3/4)
29/32            FS((3/4B2)t)                  FS(3/4B2)
15/16            FS((4/5B1)t)                  FS(4/5)
31/32            FS((5/6B1)t)                  FS(5/6)
 1/1             FS(0)                         FS(0)

From mu-ency: the Encyclopedia of the Mandelbrot Set, Copyright © 1987-2001 Robert Munafo. Robert Munafo is mrob27.

Unique Google search: mu-ency external fraction seen

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