Given an infinitive sequence 
with associative array 
, then 
is said to be a fractal sequence
1. If 
,
then there exists 
such that 
,
2. If 
,
then, for every 
,
there is exactly one 
such that 
.
(As 
and 
range through 
,
the array 
,
called the associative array of 
, ranges through all of 
.) An example of a fractal sequence is 1, 1, 1, 1, 2, 1, 2,
1, 3, 2, 1, 3, 2, 1, 3, ....
If 
is a fractal sequence, then the associated array is an interspersion.
If 
is a fractal sequence, then the upper-trimmed
subsequence is given by 
, and the lower-trimmed
subsequence 
is another fractal sequence. The signature
of an irrational number is a fractal sequence.
