I wrote up a whole few paragraphs and decided most people wouldnt be interested in the nerd stuff, woops. The short of it is that this circle diagram plots points based on fractions that get converted to rotation angle degrees (a faery diagram). What were doing is similarly to things like newtons method, plotting the iterative steps to convergence given a starting value, and comparing the iterative steps as a way to track evolution of the iteration. The way I have it set up is that each level is a plotting of a part of the continuous fraction that eventually builds to the true irrational. Every time chart color dots change is new level of the rational approximation.
Every computable real number has a unique fingerprint in the form of these spiralgraph type patterns and reading these patterns tells you something about them, in particular how hard they are to approximate to an arbitrary precision range.
Irrational numbers resist making these kinds of stable periodic orbits due to not being clean ratio fractions. By definition they cant be represented cleanly a/b .This is a gradient though because of something called continued fractional representation. Essentially while you cant represent an irrational exactly, you can find very good rational approximations that get exceptionally close to the real irrational while still remaining finitely bounded and compuable though there will always be error bounds.
Continued fractions are the algorithm that allow you to find these close-approxiations at ever increasing precision. An interesting thing is that some irrationals only need a few steps of continued fractions to reach an incredibly good approximation and some resist approximation much more. So irrationals can be catagorized by which ones quickly 'converge; to a precise approximate ratio and which ones ‘resist’ convergence.
This is the continued expansion fraction for pi. Each level in the diagram is a cutoff of that fraction structure building up to the real value. Notice that 292? That indicates a point where PI is EXCEPTIONALLY well approximated by the ratio, in this case I think its 103993/33102 any further computational effort spent at precision would be a waste.
Fractional ratio series for pi: [(3, 1), (22, 7), (333, 106), (355, 113), (103993, 33102), (104348, 33215), (208341, 66317), (312689, 99532)]…
You can see Irrationals that quickly converge to quasi-periodic orbits meaning that they approach an approximate ratio very quickly include louiville numbers and PI. Notice how they quckly converge into clumping and plotting over the same points. This is the definition of periodicity. They never quite hit the same point but its very very arbitrarily close. Pi is an example of that.
Another:
These have a distinct skinny torus wireframe shape that hugs the interior of the circle.
Meanwhile, most irrationals resist approximation much more. However after enough continued fractions assuming large enough computational resources available and microstates to search through you will find a decent rational approximation.
The golden ratio represents a very unique irrational. It is the one that resist approximation the most. It is maximally aperiodic, the most resistant possible angle value to converging on a good approximate ratio. Each step of the continued fraction is 1,1,1,1,1,.
Structural encoding wise the convergent rationals of phi (fibbonacci numbers) provides exactly the bare minimum amount of information possible to make a meaningful step towards convergence but no more. It maintains the maximal microstate entropy per informational step of computational precision gained. Theres some other interesting properties too im in the middle of formalizing.
As such, points plotted using the golden angle will essentially never overlap. They cover the circle uniformally, randomly, and close the gaps on average the most efficently. The values in the top left diagram like gap width calculate exactly how clumped the points or how uniformily they spread across the circle.
Notice though, that there are still structural patterns formed in the lines that phi traces.
!
I’m wondering if these line patterns encode the angles of a pentagon much the same way that show up in the morie pattern of penrose tilings.
Nice diagram, but could someone actually explain what we are seeing?
I wrote up a whole few paragraphs and decided most people wouldnt be interested in the nerd stuff, woops. The short of it is that this circle diagram plots points based on fractions that get converted to rotation angle degrees (a faery diagram). What were doing is similarly to things like newtons method, plotting the iterative steps to convergence given a starting value, and comparing the iterative steps as a way to track evolution of the iteration. The way I have it set up is that each level is a plotting of a part of the continuous fraction that eventually builds to the true irrational. Every time chart color dots change is new level of the rational approximation.
Every computable real number has a unique fingerprint in the form of these spiralgraph type patterns and reading these patterns tells you something about them, in particular how hard they are to approximate to an arbitrary precision range.
Rational numbers/degrees will create simple closed periodic loops tracing geometric shapes.
Irrational numbers resist making these kinds of stable periodic orbits due to not being clean ratio fractions. By definition they cant be represented cleanly a/b .This is a gradient though because of something called continued fractional representation. Essentially while you cant represent an irrational exactly, you can find very good rational approximations that get exceptionally close to the real irrational while still remaining finitely bounded and compuable though there will always be error bounds.
Continued fractions are the algorithm that allow you to find these close-approxiations at ever increasing precision. An interesting thing is that some irrationals only need a few steps of continued fractions to reach an incredibly good approximation and some resist approximation much more. So irrationals can be catagorized by which ones quickly 'converge; to a precise approximate ratio and which ones ‘resist’ convergence.
https://en.wikipedia.org/wiki/Continued_fraction
This is the continued expansion fraction for pi. Each level in the diagram is a cutoff of that fraction structure building up to the real value. Notice that 292? That indicates a point where PI is EXCEPTIONALLY well approximated by the ratio, in this case I think its 103993/33102 any further computational effort spent at precision would be a waste.
Fractional ratio series for pi: [(3, 1), (22, 7), (333, 106), (355, 113), (103993, 33102), (104348, 33215), (208341, 66317), (312689, 99532)]…
You can see Irrationals that quickly converge to quasi-periodic orbits meaning that they approach an approximate ratio very quickly include louiville numbers and PI. Notice how they quckly converge into clumping and plotting over the same points. This is the definition of periodicity. They never quite hit the same point but its very very arbitrarily close. Pi is an example of that.
Another:
These have a distinct skinny torus wireframe shape that hugs the interior of the circle.
Meanwhile, most irrationals resist approximation much more. However after enough continued fractions assuming large enough computational resources available and microstates to search through you will find a decent rational approximation.
The golden ratio represents a very unique irrational. It is the one that resist approximation the most. It is maximally aperiodic, the most resistant possible angle value to converging on a good approximate ratio. Each step of the continued fraction is 1,1,1,1,1,.
Structural encoding wise the convergent rationals of phi (fibbonacci numbers) provides exactly the bare minimum amount of information possible to make a meaningful step towards convergence but no more. It maintains the maximal microstate entropy per informational step of computational precision gained. Theres some other interesting properties too im in the middle of formalizing.
As such, points plotted using the golden angle will essentially never overlap. They cover the circle uniformally, randomly, and close the gaps on average the most efficently. The values in the top left diagram like gap width calculate exactly how clumped the points or how uniformily they spread across the circle.
Notice though, that there are still structural patterns formed in the lines that phi traces.
I’m wondering if these line patterns encode the angles of a pentagon much the same way that show up in the morie pattern of penrose tilings.
OK, i’ll have to re-read this when I’m awake… At least it starts to make sense.
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