Smaller Neighbor
Robert P. Munafo, 2002 Apr 16.
A smaller neighbor of a mu-atom is a neighbor that is smaller (has higher period). There are aleph-0 smaller neighbors in each direction (clockwise and counterclockwise). (By contrast, there are only two larger neighbors in each direction).
Example: R2.2/5a is a mu-atom. In the clockwise direction its smaller neighbors are R2.3/8a, R2.4/11a, R2.5/14a, R2.6/17a, etc. In the counterclockwise direction its smaller neighbors are R2.3/7a, R2.4/9a, R2.5/11a, R2.6/13a, etc. A few of these are seen in the figure below.
More examples:
various mu-atoms, some of which are neighbors
Near the center of the figure is R2.3/8a. To the left (the counterclockwise direction) we see two of its neighbors, R2.2/5a and R2.5/13a. The smaller one of these, R2.5/13a, is a "smaller neighbor" of R2.3/8a because it is a neighbor and it is smaller.
Similarly, to the right (the clockwise direction) of R2.3/8a we find R2.4/11a and R2.1/3a. The first of these is a smaller neighbor and the other is a larger neighbor.
See also inner neighbor, larger neighbor, Farey addition.
revisions: 19980107 oldest on record; 20120416 add illustration
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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