The Fibonacci stream

The Fibonacci stream FIB is defined by

FIB = f(FIB)

for the stream function f defined by
f(0:s) = 0:1:f(s)
f(1:s) = 0:f(s)

Another way to define FIB is first to define FIB(0) = 1, FIB(1) = 0, FIB(i+2) = FIB(i+1) FIB(i), for i =0,1,2,..., showing the relationship with Fibonacci numbers. As FIB(i+1) is a prefix of FIB(i+2) for every i, one can take the limit, which is FIB

This stream shows up remarkable patterns
Below you find a few turtle visualizations all of this single stream FIB. More precisely, a number N and two angles a0 and a1 are specified, and then elements of the stream are traversed: if the symbol 0 is read then the drawing direction is moved a0 degrees to the right; if the symbol 1 is read then the drawing direction is moved a1 degrees to the left. In both cases after doing so a line of unit length is drawn. This is repeated N times.

N = 10,800, a0 = 120, a1 = 150

N = 160,000, a0 = 144, a1 = 72

N = 500, a0 = 144, a1 = 72

N = 50,000, a0 = 30, a1 = 150

N = 39,000, a0 = 90, a1 = 120

N = 100,000, a0 = 135, a1 = 45

N = 18,000, a0 = 20, a1 = 160

N = 4,150, a0 = 18, a1 = 162

The next picture shows up colors, defined as follows. Initially the color is mainly red. This color changes smoothly until it is blue at the end. In the mean time the green part of the color varies.
N = 100,000, a0 = a1 = 174