## Chapter 5.2:

## The Curves on the

## Stopping Times Graph

So through all that we have examined, recall that the original pursuit of ours was to chase after the patterns on the Stopping Times and Max Value Graphs as far as we could. Let's return now all the way back to where we started to the x3+1 Stopping Times Graph and take a closer look.

Ok... so of all the crazy things happening in this image, I want to focus in on one particular aspect. Do you see the gently downward sloping curves all over the image? One could very reasonably ask the question: What are those curves? Are they a hint of some kind of underlying structure? Are they some important aspect of Collatz Space that we have not yet considered?

Another valid question would be in which direction are these curves oriented in anyway?

Is the pattern this way?

Maybe even this way?

Perhaps more than anything, the question I used to think about all the time was what is the shape of these curves? Logarithmic? Hyperbolic? Maybe even somehow parabolic? I used to dream of taking some scientific data-analysis software, plugging this data in, and finding a "curve of best fit" to try and study these curves....

But then one day, I didn't need to anymore.

I figured it out!