Chapter 5.5:

Studying the Divergent Entries

             Let's take a step even further into the unknown.  Let's go into totally uncharted territory and explore something that no one has ever seen before. Let's study the ones that diverge.

Consider the x5+1 world. 

             

Zoom in a little on just the left side of the Stopping Times Graph. 

             Those pink dots at the top -- those are the ones that diverge. Or at least the ones that we suspect diverge (we have the divergence cutoff set to 10^100 for this image). By this point in my research I had spent copious time looking at graphs that contained divergence, but the pink dots at the top were always slightly mysterious. For all my studying of the convergent entries, I knew almost nothing about the divergent ones at all! 

             A renegade thought began to creep through my imagination. Since the characteristic trait of Collatz Space is that it is filled with impossibly gorgeous patterns that seemingly no human will ever understand, was it possible that there were also patterns hidden out there amongst the divergent entries? I began to really want to find out. 

             I spent a few days building many machines designed to specifically examine the divergent entries. I'm going to show you the highlights, but first: Visually the results are more interesting looking if we go grab ourselves a more divergent world than x5+1. Let's use the x19+1 world. 

             The x19+1 Collatz World appears to be entirely divergent. Based on these images we're left entirely in the dark. We basically know nothing about what's happening in x19+1 space. 

             Ok. So the divergence cutoff is currently at 10^100. But here's a question: For each input n value, how many steps did it take for that hailstone to rise up and cross above 10^100? Let's plot it!

             The result is strange and slightly disconcerting. It's very hard to make any sense of. Are we looking at a pattern of some sort? I don't have too much more to tell you about this one, except that I do know that if you logarithm the x-axis, we do still see the familiar tiling phenomenon reveal itself:

             Overall? I was less than brilliantly impressed... That is, until I tried the next experiment. This time let's take each hailstone, wait until the moment it passes the divergence cutoff, and then record exactly which value it lands on; i.e. what's the first value it hits over 10^100? This is a little like the Track and Field event the long jump. Graphing this gives you:

             This, on the other hand, is quite the pattern! That's really interesting looking. In its way it looks a little tiny bit like the standard Max Value Graphs, in that it is filled with these mysterious diagonal lines, but unlike the Max Value Graphs these diagonal lines are thick and "chunky". The hailstones seem to leap over the 10^100 divergence cutoff and land into a very specific few "tracks" in some organized fashion.

 

             This pattern appears no matter what divergence-containing landscape you choose and no matter what divergence cutoff you choose! It's really quite interesting. While I've never quite figured out exactly what is causing the pattern, it is worthy of note how similar these graphs are to the Max Value graphs when both are placed in logarithmic form:

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