That concept can be extended to infinite sets S represented by line segment arrangements that satisfy two rules:
Any finite or infinite collection of line segments in which there is no ambiguity in any pair of line segments that one is overlined by the other or not is a form. From every form S we can obtain by the operation called overlining, and for every two forms F1 and F2 we can define their product (or juxtaposition) as 12. For example, an infinite collection of forms can be created as
Some interesting infinite forms can be created by recursive systems of equations. For example, a successive use of rules gives
The simplest recursive form F=, beginning from F= - , results in the sequence of natural numbers , ¼ The other recursive form F= can be called Fibonacci form because the number of line segments at depth n is nth Fibonacci number.
If F(n) denotes the number of line segments (or number of nodes in the corresponding rooted tree) at depth n of the form F, for any two forms F1 and F2 holds:
For the Fibonacci form F=, F(0)=F(1)=1 and F(n)=F(n-2)+F(n-1). Defining
the growth rate as m(F)=limn® ¥[(F(n+1))/F(n)], we obtain that the growth rate of the
Fibonacci form is the golden ratio j = [(Ö5-1)/2]