The Collatz A-B Plane
Let us now define the following: Given any two non-negative real values a ≥ 0 and b ≥ 0, associate with them their corresponding xa+b Collatz World. Assembled all together, these (a,b) points give us the "Collatz A-B Plane". Each (a,b) point on the Collatz A-B Plane corresponds to an entire xa+b world with its own set of properties.
For the sake of clarity, let me reiterate that the xa+b Collatz world is the dynamical system
When one is handed such a xa+b world, the first properties we're usually interested in are:
• What are the convergent loops in that world?
i.e. what are the loop seeds?
• Is the world entirely convergent? Entirely divergent? A mixture of both?
• And what does the imagery of that world look like?
i.e. the Stopping Times and Max Value Graphs
Let's now put this Collatz A-B Plane to some good use!
Back from the very start, the reason I invented this new method of defining non-integer Collatz realms was to investigate when and how divergence first appears between x3+1 and x5+1. Let's take a look at the worlds along that path. Since we're going through the familiar (3,1) point on the A-B Plane but changing the value of a in xa+1 as we go, I came to call this route "Path A".
Let's first do a cursory overview, checking from x0+1 to x10+1 in intervals of /\=0.1. Some important technical notes are: In each world we'll be examining exactly the first 10000 input n values. And currently the divergence cutoff is set to 10^100. If a hailstone rises above that we will, for now, classify it as divergent. Click here to open a new window with the results:
Alright... what are we looking at here?...
• From x0.0+1 to x0.9+1 we have a single convergent loop_seed at 0.
• From x1.0+1 to x1.6+1 we have a single convergent loop_seed at 1.
• At x1.7+1 it starts to crescendo up, adding successively more and more loop seeds.
• At x2.0+1 the system blows up entirely and we see that suddenly every odd integer
is its own loop_seed. A little consideration and the reason for this becomes
apparent. Take n=37. 37 is odd, so we multiply by 2 and add 1 (giving 75),
but then we round down to the nearest even number (74), and now we
divide by 2, getting us right back to 37.
We should expect the same type of behavior at x4.0+1, x8.0+1, etc...
• From x2.1+1 to x2.2+1 we seem to be "climbing down the other side" of this weird little
• But then at x2.3+1, we suddenly have an additional unexpected little loop_seed at
• From x2.4+1 to x3.5+1, these systems all have a loop_seed at n=1, but they intermittently
seem to all of their own accord have weird little other loops that come and
go. I, myself, do not see much of a pattern.
• At x3.6+1 we (shockingly?) have found the first world so far which contains divergence!
x3.6+1 is a fairly low value, not that far from x3+1. Fascinating.
• At x4.0+1 it blow up again, as expected.
• From x4.1+1 to x5.9+1 it resumes its "choppy" behavior, having one stable loop_seed at
n=1, but then other loops seem to come and go as they please.
• At x6.0+1 we spot the first world so far which is seemingly entirely divergent.
There are no convergent loops in that world at all.
• Between x6.1+1 and x6.9+1, the Collatz systems weave back and forth between being
entirely divergent and having some occasional little convergent loops.
• At x7.0+1 the Collatz Worlds crescendo up again with another massive pyramid, spiking
at the predicted peak of x8.0+1.
• And finally by x9.0+1 these Collatz systems have so much "energy" they seem to fling
all their hailstones to infinity with ease. We find all divergence.
Now that we've done a cursory overview, let's zoom in with a fine-tooth comb and see if we can learn anything about what's really happening! The following link will download a .txt file displaying info on every world from x0+1 to x80+1 in intervals of /\=0.001. (It's too long to actually post on the website directly.) Don't try and actually read all of this, it's massive. Just skim around and try and see what you can learn. Do you notice any interesting patterns or phenomena?
Really do take some time and wander around out there. It's a fascinating weird landscape. When you're satisfied, let me take you on a slight bit of a guided tour and look at some particular locations of interest.
• Skip ahead to x2.0+1. This pyramid structure is huge! Look at the sheer size of this thing.
It's very cool looking.
• Examine x2.828+1. This is very unexpected. We seem most definitely to be looking at
another strange little pyramid, but this time it has nothing whatsoever
to do with a power of 2. Really interesting.
• Examine x2.972+1. We have here an even smaller strange little pyramid. Honestly, these
things seem to dot the landscape all over the place. What are they, and why
do they exist? Pleasantly enough, I will be able to explain some of this to you
later on on this website.
• We see another big one at x3.175+1.
• The first world with divergence is x3.6+1. Very odd. The next one that appears to have
divergence is x3.984+1, followed by x3.992+1.
• The first world entirely made of divergence is x5.837+1. We see a gentle transition
towards divergence as we go to higher and higher values of a in xa+1.
• "Drive" from x12.4+1 forward. You seem to be on an empty desert highway, surrounded
by nothing but divergence on all sides. But once you've driven all the way to
x12.699+1, we see that out of the mists another of these mysterious little
pyramids appears. When you've passed the pyramid, it goes back to
being all divergence. Very very odd and interesting.
• An extreme example of this is the pyramid at x45.255+1.
• Lastly, note that immediately before the famous x3.0+1 world, we have
x2.999+1 (with loop_seeds at 1, 5, and 17)
and x3.001+1 (with loop_seeds at 1 and 103)
Does this tell us anything about the x3+1 world?
Let us now examine this "Path A" from x0+1 to x80+1 in a radically different way.