Gleichung:f=  \frac{ f^3 }{ 1-f}  \Rightarrow f^2+f-1=0 \;\;-hat \;eine\; positive\; \fgcolor{ff0000}{goldene}\; Lösung:

f=f^3+f^4+f^5+...=f^3(1+f+f^2+...)\rightarrow \frac{ f^3 }{ 1-f} 

\fgcolor{ff0000}{f}= \frac{\sqrt[2]{ 5} -1 }{2 } =0.618033988749895

the equation has a positive golden solution

f

f^2

f^3

f^4

f^5

f^6

f^7

f^8

f^9

f^{10}

\copyright \;Roman\; Chijner,\;  Heidelberg,\;  2013

n=

Eigenschaften von Baum-Fraktalen mit den Goldenen Schnitt
Properties of  Fractal Trees with the Golden Ratio

f^2 ↣

f^3 ↣

f

f^4 ↣

f^5 ↣

↣

↣

↣

↣

↣

Proj_\overrightarrow{u}:

$ \overrightarrow{u} $

Die Teile der Baum-Fraktale haben keine Überlappungen,wenn 
das Verhältnis zwischen den Längen der Zweige in benachbarten 
Ebenen dem  goldenen Schnitt  entspricht. Es gibt hier nur eine 
asymptotische Näherung der beiden Hälfte des Fractals .

The parts of the fractal tree have no overlap when the relationship
between the lengths of the branches in adjacent planes corresponds to the
Golden Section. There are only asymptotical approaches of the two
halves of the  fractal.

f = 0.618033988749895 (slider from 0 to 2, step 0.1)
C = (0.0015049395920258045, -1.0059268777200217)
D = (1.3654049841280635E-4, -0.08077481519652398)
l1 = W1(C, D, f)
l2 = Sequence(W1[Element[l1, 1], Element[l1, i], f], i, 2, 3)
l31 = Join(Sequence[Sequence[W1[Element[Element[l2, j], 1], Element[Element[l2, j], i], f], i, 2, 3], j, 1, 2])
l41 = Join(Sequence[Sequence[W1[Element[Element[l31, j], 1], Element[Element[l31, j], i], f], i, 2, 3], j, 1, 4])
l51 = Join(Sequence[Sequence[W1[Element[Element[l41, j], 1], Element[Element[l41, j], i], f], i, 2, 3], j, 1, 8])
l61 = Join(Sequence[Sequence[W1[Element[Element[l51, j], 1], Element[Element[l51, j], i], f], i, 2, 3], j, 1, 16])
l71 = Join(Sequence[Sequence[W1[Element[Element[l61, j], 1], Element[Element[l61, j], i], f], i, 2, 3], j, 1, 32])
l81 = Join(Sequence[Sequence[W1[Element[Element[l71, j], 1], Element[Element[l71, j], i], f], i, 2, 3], j, 1, 64])
s1 = Sequence(Segment[Element[l1, 1], Element[l1, i]], i, 1, 3)
a = Segment(D, C)
s2 = Join(Sequence[Sequence[Segment[Element[Element[l2, j], 1], Element[Element[l2, j], i]], i, 2, 3], j, 1, 2])
s3 = Join(Sequence[Sequence[Segment[Element[Element[l31, j], 1], Element[Element[l31, j], i]], i, 2, 3], j, 1, 4])
s4 = Join(Sequence[Sequence[Segment[Element[Element[l41, j], 1], Element[Element[l41, j], i]], i, 2, 3], j, 1, 8])
s5 = Join(Sequence[Sequence[Segment[Element[Element[l51, j], 1], Element[Element[l51, j], i]], i, 2, 3], j, 1, 16])
s6 = Join(Sequence[Sequence[Segment[Element[Element[l61, j], 1], Element[Element[l61, j], i]], i, 2, 3], j, 1, 32])
s7 = Join(Sequence[Sequence[Segment[Element[Element[l71, j], 1], Element[Element[l71, j], i]], i, 2, 3], j, 1, 64])
s8 = Join(Sequence[Sequence[Segment[Element[Element[l81, j], 1], Element[Element[l81, j], i]], i, 2, 3], j, 1, 128])
l91 = Join(Sequence[Sequence[W1[Element[Element[l81, j], 1], Element[Element[l81, j], i], f], i, 2, 3], j, 1, 128])
l101 = Join(Sequence[Sequence[W1[Element[Element[l91, j], 1], Element[Element[l91, j], i], f], i, 2, 3], j, 1, 256])
s9 = Join(Sequence[Sequence[Segment[Element[Element[l91, j], 1], Element[Element[l91, j], i]], i, 2, 3], j, 1, 256])
s10 = Join(Sequence[Sequence[Segment[Element[Element[l101, j], 1], Element[Element[l101, j], i]], i, 2, 3], j, 1, 512])
b = OrthogonalLine(D, a)
A = Point(s1)
B = Point(b)
α = Angle(A, D, B)
c = Line(D, C)
d = OrthogonalLine(B, b)
n = 3 (slider from 1 to 10, step 1)
Textfeld1 = Textfield(n)
r02 = Element(Element[l31, 2], 1)
r03 = Element(Element[l41, 4], 1)
r01 = Element(Element[l2, 1], 1)
r04 = Element(Element[l51, 8], 1)
r05 = Element(Element[l61, 15], 1)
r06 = Element(Element[l71, 30], 1)
r07 = Element(Element[l81, 59], 1)
r08 = Element(Element[l91, 118], 1)
r09 = Element(Element[l101, 235], 1)
r10 = Element(Element[l101, 235], 3)
R01: (r01 + D) / 2
R02: (r01 + r02) / 2
R03: (r02 + r03) / 2
E = Point(b)
u = Vector(D, E)
Text5: "" + (LaTeX["f*cos(α)=" + (TableText[{{Sum[Sequence[LaTeX[If[i ≟ 3, "", "+"]] + "f^{" + (LaTeX[i]) + "}cos(α)" + (LaTeX[If[i ≟ 10, "+...=>", ""]]), i, 3, n]]}}])]) + ""

Tool: W1
l(B,C,D)
Input: A, B, f
Output: b0

C1: B + Rotate[B - A, 60*°] * f
D1: B + Rotate[B - A, 300*°] * f
b0: {B, C1, D1}