| Secondary Continental Mu-Atom |
Robert P. Munafo, 2003 Sep 27.
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 = 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(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 LM=0 where LM if the Mth 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.
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.
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