Chapter 6.3:

The Wiggle

             Let's take a precise look at exactly what this "wiggle" looks like. Let's use those same 6 branches from x3+1 space as our case study. Graphing the % memberships out to 10^5 gives these images. The smaller pictures to the right are zoom-ins. 

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Graphing further to 10^6 gives us:

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Graphing further to 10^7 gives us:

Screen Shot 2020-04-25 at 1.14.18 AM.png

Zooming in on just the tail end of

the pink wiggle (the branch at n=80) reveals that

it really is starting to be very very stable: 

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Graphing further to 10^8 gives us:

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And finally graphing even further to 10^9

(which takes a long time to process!) gives us:

Screen Shot 2020-04-25 at 1.26.10 AM.png

Having explored all the way out to 10^9, we can now do a close zoom-in on the

green one (the branch at n=512) reveal the true nature of the sinusoidal wiggle:

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             These wiggles appear to occur as the long-term % membership of each and every non-trivial branch in x3+1 space. Perhaps the take away from all of this is the following key observation:

Conjecture: In x3+1 space every non-trivial branch's

% membership function %(n) has LimSup > 0. 

             Another way to state this is to say that all non-trivial branches of x3+1 space "hold their own"

as n goes to infinity and never vanish down to a % membership of 0. 

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