# Neighboring Fraction

Farey neighbors
Robert P. Munafo, 2012 Apr 18.

Two fractions a/b and c/d are "neighboring fractions" (or "Farey neighbors") if and only if either of the following (equivalent) statements is true:

ad - bc = {+-}1

|a/b - c/d| = 1/bd

If the two neighboring fractions are between 0 and 1, and are both in reduced form, then the mu-atoms with these fractions as their internal angles are neighbors. Their inner neighbor will be the mu-atom with the angle given by Farey addition of a/b + c/d, which is the angle (a+c)/(b+d).

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 neighboring fractions. There is a table giving lots of additional examples of pairs of neighboring fractions in the secondary continental mu-atoms article.

Sources

Robert Devaney, The Mandelbrot set and the Farey tree, 1997.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017. Mu-ency index

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2017 Feb 02. s.11