Chapter 5.7: Dissecting the Stopping Times Graph, One Branch at a Time

             One day, whilst staring at the Tree Diagram and the Stopping Times Diagram:

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             I began to wonder what would happen if you selected only one one particular branch on the Tree Diagram (one of the lower down ones maybe) and graphed just those points on the Stopping Times Graph — what would the resultant image look like? For this purpose I designed another Python program. Let me show you how it works:

             So this particular branch we were examining in the above video. Let's take a closer look at the result. We're looking at just the branch at n=320:

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             Playing with this program, I accidentally discovered that the further deep down into the Collatz Tree you explore ... a funny thing starts to happen. You see the image on the left there? If I might ask, where did the funny little horizontal bands in the Stopping Times Graph go?

 

             When you look farther and farther out into the tree, the little bands seemingly vanish altogether and there is only a smooth perfect grid of dots. (These "dots", if you zoom in on them, are actually still tiny little horizontal bands... but the further down the tree you look, the more and more tiny the little bands become.) Why is this? 

 

             I thought a lot about this. Let me explain to you what I figured out...