Chapter 5.8:

The Secret of the Stopping Times Bands

             Consider the image to the right. When going from a lower dot to its "parent dots" above there are only two kinds of steps we can take. A step up to the right is an 2*n step. A step up to the left is a (n-1)/3 step. Every dot has a "right parent". But only those dots which are 4 (mod6) have a "left parent". 

             Consider the node at n=16. I'm going to sketch out two paths from 16 to get higher into the tree. Starting at n=16 we can take a left (n-1)/3 step, taking us to n=5, and then take two right 2*n steps taking us to n=10, then n=20. OR we could start at n=16 and take two right 2*n steps taking us to n=64, and then take a left (n-1)/3 step arriving at n=21. ..... n=20 and n=21 form the very fledgling start of a horizontal band. 

             Just for a moment, suppose we were not studying the normal x3+1 Collatz Tree. Suppose we were studying some kind of "idealized" Collatz Space where steps to the right were still 2*n, but now steps to the left are actually just n/3 instead of (n-1)/3. We see that, starting at n=16, taking one step left and two to the right OR taking two steps right and one left now result in the exact same position, namely 21.333. 

             In some ways in the image above n=20 and n=21 "want" to coincide on the same point, but they cannot due to the -1 in (n-1)/3. 

The –1 takes what otherwise would be a perfectly "idealized" Collatz Stopping Times Graph image, as we see in the lower picture, and effectively fractures all of the dots into a small horizontal band, separated from each other only by exactly when and how they took their (n-1)/3 steps. Each horizontal band consists of exactly those hailstones which (on their descent to n=1) take the same number of right steps as each other and the same number of left steps as well. 

             In so saying, we finally have a solid understanding of why the stopping times graph has the famous little horizontal bands! Many professional math papers have been submitted in attempts to explain them... but we've finally got it! The Stopping Times Graph literally IS the Tree Diagram, but what would be its perfect "idealized" grid-like appearance is fractured into horizontal bands by repeated applications of -1 steps. 

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