External Angle
Robert P. Munafo, 2003 Dec 18.
A one-to-one mapping of the unit circle onto the boundary of the Mandelbrot set which yields a simple method of naming any point on the boundary. The external angle, also called external argument, is a real number between 0 and 1, equal to the angle on the unit circle which has been mapped divided by 2{pi}.
There are many (2c) 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. The binary decomposition entry shows how it is computed.
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, tips, etc.
The external angles with powers of 2 in the denominators correspond to the bifurcation in rotational symmetry of the filaments surrounding all islands. This also shows up in the symmetry of embedded Julia sets and their paramecia.
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.
This illustration (based on the image of R2F(1/2B1)FS[2]FS[2]FS[2]S) shows the "first" 32 external angles.
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Here is a table giving the corresponding FS operators in the R2 Naming System:
| 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 the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index
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