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Robert P. Munafo, 2023 Mar 22.

The term "larger" is sometimes used in this encyclopedia to refer to a relationship between mu-atoms that are siblings: A mu-atom is "larger" if it has a lower period. The word is used in this way on the pages: binary search for internal angle, internal angle, larger neighbor, neighbors, and secondary continental mu-atom. The "larger neighbor" term is also used elsewhere, e.g. Farey addition. See also smaller.

R2.1/3a is ''larger'' than R2.2/5a because it has a lower period
R2.1/3a is "larger" than R2.2/5a because it has a lower period

For more examples of larger neighbors, see the table in the secondary continental mu-atom article.

Being "larger" in this sense does not necessarily mean that a mu-atom is physically larger: for example, R2.3/8a is physically larger than R2.1/7a, but is smaller in the sense meant here, because it has a higher period.


In the images above, both mu-units are shown at the same scale (image width and height are 0.1 units on the real and imaginary axis respectively). But the mu-atom with the lower period is also physically smaller. This is a counter-example to the ordinary rule that siblings with a lower period are physically larger. The discrepancy is accounted for by the sin(2π N/M) term in the "Milnor's approximation" formula shown in the secondary continental mu-atom article.

Some Things That Are "Largest"

The largest mu-atom is the entire Mandelbrot set itself, but commonly the term is used to refer to the largest secondary continental mu-atom, which is R2.1/2a. Likewise, the largest mu-unit (aside from the entire Mandelbrot set treated as a mu-unit) is R2.1/2.

The largest mu-molecule is also the entire Mandelbrot set, and the largest island mu-molecule is R2F(1/2B1)S.

revisions: 20120416 oldest on record; 20230322 add "some things..." section

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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