Secondary Continental Mu-Atom  

Robert P. Munafo, 2012 Apr 16.



The secondary continental mu-atoms are the mu-atoms that touch the main cardioid. (See also secondary and continental)

These mu-atoms are usually the first pattern that people notice when they see the Mandelbrot set for the first time. As a result, it is common to discover the patterns within these mu-atoms and develop that knowledge into an Analytical Naming System.

Contents

Precise Location Formulas

Examples

Full Listing up to Period 16


Precise Location Formulas

The position of the bond point between R2a and any secondary continental mu-atom R2.N/Ma is:

a2 - b2 + 1/4 + 2ab i

where

a = sin(2 pi N / M) / 2
b = (1 - cos(2 pi N / M)) / 2

and N/M is the mu-atom's internal angle. Expressed as a complex number, the bond point is located at:

B = lambda (2-lambda) / 4

where

lambda = e2 pi N/M

is a point on the unit circle. The derivative of B with respect to lambda is:

dB = 1/2 (1 - lambda) dlambda

the radius of the mu-atom R2.N/Ma is the absolute value of this divided by M2, which is equivalent to this approximation given by John Milnor:

radius ≅ sin(2π N/M) / M2

The position of R2.N/Ma's nucleus is approximately

B - i dB / M2

but this is only precise in the case of R2.1/2a. For all others, it can be used as an initial value for the Newton-Raphson method, since the nucleus is a solution to the equation LM=0 where LM is the Mth Lemniscate.

These formulas can be generalized; see mu-atom size formulas.


Examples


A part of the continent with some secondary and tertiary mu-atoms labeled
A part of the continent with some secondary and tertiary mu-atoms labeled


In the figure above, the following mu-atoms are secondary continental: R2.5/12a, R2.2/5a, R2.5/13a, R2.3/8a, R2.4/11a, and R2.1/3a.

The mu-atom R2a is the continent seed or "main cardioid"; the other labeled mu-atoms (such as R2.1/3.1/2a) are tertiary continental mu-atoms.


Full Listing up to Period 16

The following table lists all secondary continental mu-atoms with denominators less than or equal to 16. It is useful for finding mu-atoms given their R2 names.

For example, suppose you want to locate R2.3/8a. Its internal angle is 3/8; using the table, you find that the internal angles of its larger neighbors are 1/3 and 2/5. This means that R2.3/8a is located between R2.1/3a and R2.2/5a. If you do not know where these mu-atoms are located, you could use the table again to locate their larger neighbors.

Internal CW Larger CCW larger Internal CW Larger CCW larger Angle Neighbor Neighbor Angle Neighbor Neighbor   1/2 --- --- 5/9 1/2 4/7 1/3 --- 1/2 5/11 4/9 1/2 1/4 --- 1/3 5/12 2/5 3/7 1/5 --- 1/4 5/13 3/8 2/5 1/6 --- 1/5 5/14 1/3 4/11 1/7 --- 1/6 5/16 4/13 1/3 1/8 --- 1/7 6/7 5/6 --- 1/9 --- 1/8 6/11 1/2 5/9 1/10 --- 1/9 6/13 5/11 1/2 1/11 --- 1/10 7/8 6/7 --- 1/12 --- 1/11 7/9 3/4 4/5 1/13 --- 1/12 7/10 2/3 5/7 1/14 --- 1/13 7/11 5/8 2/3 1/15 --- 1/14 7/12 4/7 3/5 1/16 --- 1/15 7/13 1/2 6/11 2/3 1/2 --- 7/15 6/13 1/2 2/5 1/3 1/2 7/16 3/7 4/9 2/7 1/4 1/3 8/9 7/8 --- 2/9 1/5 1/4 8/11 5/7 3/4 2/11 1/6 1/5 8/13 3/5 5/8 2/13 1/7 1/6 8/15 1/2 7/13 2/15 1/8 1/7 9/10 8/9 --- 3/4 2/3 --- 9/11 4/5 5/6 3/5 1/2 2/3 9/13 2/3 7/10 3/7 2/5 1/2 9/14 7/11 2/3 3/8 1/3 2/5 9/16 5/9 4/7 3/10 2/7 1/3 10/11 9/10 --- 3/11 1/4 2/7 10/13 3/4 7/9 3/13 2/9 1/4 11/12 10/11 --- 3/14 1/5 2/9 11/13 5/6 6/7 3/16 2/11 1/5 11/14 7/9 4/5 4/5 3/4 --- 11/15 8/11 3/4 4/7 1/2 3/5 11/16 2/3 9/13 4/9 3/7 1/2 12/13 11/12 --- 4/11 1/3 3/8 13/14 12/13 --- 4/13 3/10 1/3 13/15 6/7 7/8 4/15 1/4 3/11 13/16 4/5 9/11 5/6 4/5 --- 14/15 13/14 --- 5/7 2/3 3/4 15/16 14/15 --- 5/8 3/5 2/3

The use of "secondary" here is not universal. For example, Prof. Devaney refers to these mu-atoms as primary bulbs. See also fixed-point cardioid.


revisions: 20030927 oldest on record; 20120416 add illustration and example of using the table




From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017.     Mu-ency index


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