That concept can be extended to infinite sets S represented by
line segment arrangements that satisfy two rules:

each arrangement S has a single uppermost line segment, and

the collection of line segments overlined by that
uppermost line segment is a disjoint union of the members of S.
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 F_{1} and F_{2} we can define their
product (or juxtaposition) as _{1}_{2}. 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 F_{1} and F_{2} holds:

(F_{1}F_{2})(n)=F_{1}(n)+F_{2}(n) and

(n)=F(n1).
For the Fibonacci form F=, F(0)=F(1)=1 and F(n)=F(n2)+F(n1). Defining
the growth rate as m(F)=lim_{n® ¥}[(F(n+1))/F(n)], we obtain that the growth rate of the
Fibonacci form is the golden ratio j = [(Ö51)/2]
