Adjacent Fractions
Robert P. Munafo, 2012 Apr 18.
Two fractions a/b and c/d are adjacent if there is an integer e such that any of the following (equivalent) statements is true:
a/b + 1/e = c/d or c/d + 1/e = a/b
a/b - c/d = 1/e or c/d - a/b = 1/e
a/b - c/d = ±1/e
|a/b - c/d| = 1/e
The converse of this rule is not true. For example, 7/30 plus 1/6 is 2/5, but R2.7/30a and R2.2/5a are not neighbors (R2.1/4a, R2.1/3a, and R2.4/17a lie between them).
Similarly, 5/12 minus 3/8 is 1/24, but R2.5/12a and R2.3/8a are not neighbors (R2.2/5a lies between them).
If two fractions are neighboring fractions, then they are also adjacent fractions, but the reverse does not hold true.
In the Mandelbrot set, if two siblings are neighbors, their internal angles are neighboring fractions (and therefore also adjacent fractions).
For example, R2.1/3a and R2.2/5a are neighbors, and the difference of the internal angles is 1/15. See the neighbors page for an illustrated example of pairs of mu-atoms that are neighbors; you can verify that for each pair of neighbors the two fractions are adjacent. There is a table giving lots of additional examples of pairs of neighbors in the secondary continental mu-atoms article.
See also between, Farey addition.
revisions: 20010123 oldest on record; 20120416 expand definition and add links; 20120418 reference neighboring fractions
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2012 Apr 19.
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