The Mandelbrot Set and Julia Sets

Combinatorics in the Mandelbrot Set - the Farey Sequence

Between n- and (n+1)-cycle discs of a principal series, the smallest cycle number is 2n+1:
between the 2 and 3 is a 5
between the 3 and 4 is a 7
between the 4 and 5 is a 9
and so on.
Between consecutive discs whose cycles we have already found, the smallest cycle number is the sum of those just found. For example,
between the 2 and 5 is a 7
between the 5 and 3 is an 8
between the 3 and 7 is a 10
between the 7 and 4 is an 11
between the 4 and 9 is a 13
between the 9 and 5 is a 14
between the 5 and 11 is a 16
between the 11 and 6 is a 17
and so on.
This last rule persists to all levels: between consecutve discs with cycle numbers p and q, the smallest cycle number is p+q. This arrangement of features is called the Farey sequence.

Return to Combinatorics in the Mandelbrot Set.