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.
Graphing further to 10^6 gives us:
Graphing further to 10^7 gives us:
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:
Graphing further to 10^8 gives us:
And finally graphing even further to 10^9
(which takes a long time to process!) gives us:
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:
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.
