Proceed to Safety

Inner Neighbor    

Robert P. Munafo, 2012 Apr 16.

Given two mu-atoms that are neighbors, the inner neighbor (also called the common smaller neighbor) is the largest mu-atom between them. "Between" in this case refers to any mu-atoms that you encounter as you travel along the boundary of the parent from one sibling to the other. (There are two ways you can go, the way you should go is the way that doesn't cross the parent's parent or the parent's cusp if the parent is a seed.)

some mu-atoms
some mu-atoms

In this figure, R2.2/5a and R2.1/3a are neighbors. The mu-atoms R2.5/13a, R2.3/8a and R2.4/11a are all said to be between them.

R2.5/13a, R2.3/8a (and smaller unlabeled ones) are neighbors of R2.2/5a. R2.3/8a, R2.4/11a (and smaller unlabeled ones) are neighbors of R2.1/3a.

The inner neighbor or common smaller neighbor of R2.2/5a and R2.1/3a is the only mu-atom that is a neighbor to both, namely R2.3/8a. It is also the largest mu-atom that is between R2.2/5a and R2.1/3a.

Additional examples of inner neighbors:

R2.2/5a and R2.3/8a are neighbors. Their inner neighbor is R2.5/13a.

R2.3/8a and R2.1/3a are neighbors. Their inner neighbor is R2.4/11a.

See also larger neighbor, smaller neighbor, Farey addition.

revisions: 20010123 oldest on record; 20120416 add illustration

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

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