This page is dedicated to being a resource for understanding the Golden Mean - PHI -
The 'golden mean' is classically defined as the proportion whose whole is to the greater part as the greater part is to the lesser.
It is generally denoted by the Greek letter PHI (Ø). Sometime it is written as the letter 'gamma' or 1/Ø
Ø = 1.61803399... and 1/Ø = 0.6180339...
Phi was the ancient ideal of beauty as embodied in perfect proportions. Phi was a central theme in classical math, architecture, and mysticism. More to the point, however, Phi is found to be everywhere evident in nature herself as the universal pattern for growth and structure. It is the prime fractal embedding scheme for nature's scaffolding. Phi is found to have prime significance in wave motion, energy transfer, and possibly even sacred spiritual matters. Mathematically it is directly related to the other famous transcendentals - pi, e, and i.
This page is structured in two parts:
I) The factual properties of Phi
II) Speculation about Phi's greater significance in Life
Period Doubling Route to Chaos
Divide a whole into a greater part P and a lesser part p such that
the whole is to the greater as the greater is to the lesser.
(P+p)/P = P/p = PHI = Ø by definition
:: Ø = P/P + p/P
:: Ø = 1 + 1/Ø
:: ز = Ø + 1
:: ز - Ø - 1 = 0
This is a simple quadratic equation with two roots
Ø = (1 ± sqrt5)/2
The positive root is Ø = 1.6180339
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Phi is imminently self referential. This recursive relation is given by ...
PHI = 1/(PHI-1)
PHI - 1 = 1/PHI
PHI = 1 + 1/PHI
PHI*PHI = PHI + 1
:: as a power series we have
ع= 1/(Ø-1) = 1.618... = 1/.618...
ز = ØØ = (Ø + 1) = (1.618...)² = 2.618... = 1.618... + 1
س = Øز = Ø(Ø +1) = (ز + Ø) = 2.618... + 1.618... = 4.236...
Ø^4 = Øس = Ø(ز+Ø) = س + ز
...
in general Ø^n = Ø^(n-1) + Ø^(n-2)
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Take any two numbers and add them to get a third. Take that number plus the last to get a fourth and so forth ... that's a Fibonacci sequence. Each term is the sum of the two previous terms.
i.e. 1, 1, 2, 3, 5, 8, 13, 21, 39, 60, 99, 159, 258... etc.
or 12, 28, 40, 68, 108, 176, 284,...
or 3, 9, 12, 21, 33, 54, 87, 141, 228, ...
Now take the ratio of any term to its previous:
For the first sequence we have 1/1 = 1, 2/1 = 2, 3/2 = 1.5,
5/3 = 1.666... and last we have 258/159 = 1.622...
It is found that every Fibonacci sequence ratio tends toward the Golden Mean or PHI = 1.618...
For the second sequence above 284/176 = 1.613...
The third sequence has 228/141 = 1.617...
The further one expands the sequence the closer the ratio comes to PHI. Extremely interesting is the fact that the number ratios oscillate between greater and lesser than PHI. If one plots the values on an xy graph, the shape is that of a typical damped oscillation which smooths out as time goes on.
Now what happens if we start our sequence with 1 and PHI itself?
we have: 1, Ø, Ø+1, 2Ø+1, 3Ø + 2, 5Ø + 3, ...
ratio one = Ø/1 = Ø
ratio two = (Ø+1)/Ø = 1 + 1/Ø = 1 + Ø - 1 = Ø
ratio three = (2Ø + 1)/(Ø+1) = (Ø+(Ø+1))/(Ø+1) = Ø/(Ø+1) + 1 = 1 + 1/Ø = Ø
For all terms, the last divided by the previous is exactly Ø!
It is interesting to note also that for any term, it is less than 2 times the previous term and it is greater than 2 times the term before that. In fact, by definition, the term is equal to 2 times the average of the two preceeding terms. This suggests that a growth process of this sort does a doubling based on an average of its prior states. When we get to the section called Period Doubling Route to Chaos, it might be fruitful to keep this fact in mind.
The above demonstrates that a Phi based Fibonacci is perfect in and of itself from the start, but there is more. Consider a geometric or multiplicative sequence of Phi.
1, Ø^1, Ø^2, Ø^3, Ø^4, Ø^5, ... the powers of Phi
each term divided by the previous is again Phi
Now let us take the sums of terms as in a Fibonacci.
Recall that in PHI's Relations to itself
we demonstrated that in general Ø^n = Ø^(n-1) + Ø^(n-2)
This means that the geometric series of Ø is a perfect Fibonacci also!
All the above shows that the golden mean is golden because it stands in relation to itself recursively by addition as well as multiplication.
Ø = 1/(Ø-1) IS this recursive relationship. Beauty is golden, because beauty reflects itself in itself. Mirror mirror on the wall, who is the fairest of them all ...
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Take a line segment and divide it into a greater and lesser part in the ratio of Phi. Now divide the greater part by the lesser. The new segments created stand in the same ratio. This process is equivalent to flipping the smaller part onto the larger and then taking the still smaller part and flipping it back onto the smaller. Repeating this process, involves dividing each smaller segment by phi. Each step is like a reflection down a hall of mirrors. I have not been able to prove it, but the end point appears to converge upon the value 1 - 1/Ø = 2 - Ø = 0.381996...
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Make a rectangle with sides in the ratio of phi:1
Take the unit side and inside make a unit square. The small rectangle left over has sides in the proportion 1/(phi -1). But phi-1 = 1/phi , therefore the ratio of the smaller rectangle's sides is phi. This recurses to infinity, always in the ratio phi:1
This leads us to the:
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The following info (in between cut lines ) was clipped from http://www.mathsoft.com/asolve/constant/gold/gold.html
This will be removed as I work up my own material
The length and width of the nth Golden
rectangle can be written as linear expressions 
where
the coefficients a and b are always Fibonacci numbers! These
Golden rectangles can be inscribed in a logarithmic spiral as
pictured. Assume that the lower left corner of the first
rectangle is the origin of an xy-coordinate system. Question:
what is the accumulation point for the spiral? Answer:
Such logarithmic spirals are "equiangular" in the sense
that every line through 
cuts across the spiral at a constant
angle 
. In this way, logarithmic spirals generalize
ordinary circles (for which =90 deg). The logarithmic spiral pictured
gives rise to the constant angle
Logarithmic spirals are found throughout nature, for example, the
shell of a chambered nautilus, the tusks of an elephant and
patterns in sunflowers and pine cones.
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From: Swinton Roof
From: Swinton Roof
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Period Doubling Route to Chaos
Phi has been found to be involved in non-linear growth process. We will give a basic view of its involvement here. After reading this section one can go to Logistic Parabola for an in-depth discussion of how Phi mathematically emerges. Consider a linear growth process. A production or state value S is given by some rate 'R' by time 't' for example: S = Rt. If the process state, however, depends on the previous state we get a function such as Sn+1 = R Sn. Now, over time the process will grow exponentially so that Sn = S0 x R^n. Most growth is limited at some point by resources or whatever. We can incorporate this by letting the rate equal to R(1-S) such that as S approaches 1, the rate goes to zero. The self-limiting growth function then becomes S -> R(1-S)S This is the classic so-called 'logistic parabola' equation. It has been used to model rabbit populations and all sorts of non-linear processes which exhibit periodic and chaotic behavior. The function is called parabolic because S = RS - RS^2 is a quadratic equation. The curve forms an inverted parabola in the xy plane. R(1-S)S is the simplest model of non-linear growth process. It is found that such processes exhibit extraodinary behaviours with certain values of the parameters R & S.
Behavior can be fixed, periodic, or chaotic. The logistic parabola rises from zero at S=0 to a maximum at S=1/2 and back to zero at S=1. Note that this is a normalized simplified model of process. This basic inverted parabola curve is a sort of map of how the process behaves. It is a continuos model for a discrete iterative process however, so the actual process states, will jump around on this curve in steps. If we take R(1-S)S and start iterating some initial point we get what is called an 'orbit'.
An orbit is just a sequence of process states S0,S1,S2,S3,S4,...etc. where each Sn+1 = R(1-Sn)Sn. Some starting points will produce a fixed point or S0 = S1 = S2 = ... forever. This is what is called a period 1 orbit.
Other orbits oscillate between two values S0, S1, S0, S1, etc This is called a period 2 orbit. One can in general have orbits up to an infinite period, at which point one says the orbit is chaotic or random, because it never repeats itself.
Orbits which tend to bounce around some particular value are said to be attracted. The point is a fixed attractor and the orbit is said to be stable if there is a convergence toward the attractor. It is said to be indifferent if it doesn't actually converge over time. It is said to be super-stable if it converges rapidly to the fixed point.
The main factor that determines the kind of orbits or process behavior is the rate R. Certain values of R give rise to super stable fixed point attractors. As the value of R is increased however, the orbits become indifferent and eventually the fixed point bifurcates so-to-speak and the orbits circulate about two points now. One gets a period 2 orbit. These point attractors eventually bifurcate also with even greater values of R.
The process continues on with periods 4,8,16,32,...to infinity or chaos. It turns out that when R = 2 Phi = 2(1.6180339...) = 3.2360679... one gets a super-stable period two orbit. What this means is that Phi enters into non-linear process as the rate parameter which produces the first island of stability in an otherwise possibly chaotic situation. The fixed point period one situations are degenerate in a sense and uninteresting because they represent a frozen or close to equilibrium case. In living systems, for example, that would be death. The Golden Mean, however, provides the first point of dynamic stability, wherein one can have change but still remain integrated. Not far away on the spectrum is further bifurcation into mutiple period doubling right into chaotic behavior. The Golden Mean thus is on the edge between order and chaos. A system whose growth parameters are constantly being influenced will in general show a range of behaviours which may be chaotic. If it comes close enough to the Golden Mean, however it will be attracted into stabilizing itself. The system is said to be self-organizing and can reassert it's implicit integration right out of the fires of chaos.
Below is a plot of orbits for different values of R. The state values or orbits are plotted on the y axis, while R is along the x axis. To visualize the interpretation, imagine a vertical line through any particular value of R. The orbit for that particular situation is plotted as a series of points on that vertical line. The period doubling is evident as is the region on the right where chaos reigns.
Visit Logistic Parabola for an in depth derivation of R= 2Phi as a period 2 orbit.
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Speculation about Phi's greater significance in Life
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