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. a=1, b=3, c=2, and d=5. The first statement above is true because 1×5-2×3 = -1. The difference of the internal angles |1/3-2/5| 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. For example, find " 7/9 3/4 4/5 " in the table; 3/4 and 4/5 are the two larger neighbors of 7/9. They satisfy the statements above: 7×4-9×3=1 and 7×5-9×4=-1. In addition, since 7/9 is smaller than both of these larger neighbors, 3/4 and 4/5 are also each other's neighbors: 3×5-4×4=-1.

Sources

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

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