Pi and Phi are to two common features of converging series within a logarithmic spiral according to the golden ration (3/5).
Pi converges utilizing a series of increasing numbers according to the formula 1, 3, 5, 7 utilizing the formula 4/1-4/3+4/5-4/7.
Phi converges utilizing division with F series of numbers (x,a,a+a,a+a+a,(a+a+a+a+a),(a+a+a+a+a+a+a+a),etc basically adding each of the two prior numbers to get to the next number: 0,1,1,2,3,5,8,etc You'll note in the initial formulation zero is excluded in favor of x.
This can be revisited, but we're dealing with a quantum length and x represents g-space or a non-informational state in NLC. the convergent series (phi-how clever, pi to phi=fi) in this case is one number divided by the prior number:
1/x, 1/1, 2/1,3/2,f/f-1 for the entire series.
In such a system, the "quantum" size is found at the initial point in the series (having a ratio of 1/2) with a radius of 1 (therefore a length of pi*1 where 1 is the quantum length. This is a unique relationship since all other arcs are a function of r=fibrunicci(pi/2).
so you initially have:
1,1,2,3,5,8,13 etc as the F series. The first step in the series 1:1 is not seen any more than the first stop in determining pi (4/(-1)) for the simple reason that there is no length beyond 1, you cannot cut a quantum distance in half. In terms of information theory, this is one bit. Half a bit (yes/no) would be neither yes or no, instead it would be half yes and half no which I presume means it would be a "maybe". For those of you looking for randomness in NLC theory, perhaps you've found it.
No perfect circle or perfect spiral can be drawn unless drawn to this minimum size because the two are represented by converging (but never arriving).
The quantum point in an F series converges with pi for several reasons. At this minimum length the arc is a whole arc, as opposed to a half arc. It is also half a sphere with radius 1. The "area" of this arc/half circle is pi*r.
This unique relationship (a function of quantum math as much as anything) changes radically where the 3/5 ratio is established at the second step in the series from 1:2 to 2:3. which results from the shifting of the center of the arc which has a new center shifted 1 space, the next center being shifted...2 spaces, the next 3 and so on in a mirror of the series. The difference being that in the unique 1:2 space the arc originates from the center and that never happens again. Technically, this is because you have to start each line at a corner and the arc has to contact with the prior arc. At least one of you is wondering about this initial arc, and you're well advised to do so. The 1:1 and 1:2 arcs are two halves which are identical. In the language above we are having information multiply.
At least one of you is asking about the 0-1 arc and if you want to see what it doesn't exist (in two dimensional space) try drawing it, it has nowhere to go. In this case it is a quantum length so you can't divide it, so don't try. You are thinking right now "but I can cut any space in half" but this is information and if you cut it in half you only have a maybe and we have to save maybe math for later. If you are uncomfortable with this, you need to come up with your own theory of the universe, this is mine. Anyway, for the 1:1 part of the series it is the only place where the two arcs have the same point of origin and hence the 1:1 and 1:2 can be drawn without "lifting" the central pivot of the arc. The remaining arcs shift with the F series (if you don't believe me, try it).
The relationship of the two infinite series and phi is shown at this point according to this equation:
The length of the arc is equal to quantum length at this point and the gold ratio at this point is one so at the quantum length (unlike everywhere else in the universe) the length is pi*r/2, r=1, and phi is one. At this quantum length, then pi*r/2=pi*phi/2 and r=phi. Phi alters past the quantum length and approaches 1.61803399 but never ends until the far end of the spectrum where pi is solved based on the number of places out that pi goes and phi is solved by going out as many places both being a function of the total amount of information in space.
The two are related outside of quantum space by the equations: 2cos(pi/5)=phi or 2sin(pi/5)=(3-phi)^1/2 which is the mathematical relationship between the arcs described above and the movement of the center point according to the fibrunicci equation (The "pointer" of a compass starts in the center and moves one quantum space around boxes drawn according to the 3/5 (golden) ratio. 0 spaces, 1 space (from the zero space), 2 spaces (from the prior center point of one space movement), 3 spaces from the second starting point, etc so that you can draw a F "expanding quantum line, taking a 90 degree turn after each F quantum points which encompasses the curve of the spiral which are sequentially defined by r=next number in the F series and a starting point at the end of the prior corresponding F series along the quantum line.
If you draw out this first quantum 1:1 and 1:2 location you are struck with the only curve within the pantheon of curves of the gold triangle that seems to vibrate from the common starting point, as if uncertain in which direction to travel. It is almost like a fickle lover, in love one day, the next callous and cold, one day ready to love forever, the next full of scorn and disdain.
One can see at this unique point in a sprial universe, that it is the only place where curves traveling in both directions could originate and one can well imagine before the unwinding of time within this 180 degree curve, all of the event of time, all of the information moving back and forth like a metronome as linearity begins it crucial first steps as a vibrating needle with its apex at the center of the two 1x1 squares, all the yeses and nos of the fickle lover balanced in the algorithm. Yes and no, and in between maybe.
https://www.youtube.com/watch?v=8Vajq-UK2aE
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