Secondary Continental Mu-Atom

Robert P. Munafo, 2003 Sep 27.

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.

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 = e^{2 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(pi 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 Newton's Method, since the nucleus is a solution to the equation L_M=0 where L_M if the M^th Lemniscate. (See also R2.1/3a).

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

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.

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

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.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index

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