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 2
N-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