## Chapter 5.3: The Discovery of the Tiling Phenomenon

So... in pursuit of the understanding of this question, one day I was looking carefully at the graph of the x3+5 Collatz World:

And I took out a ruler and started very carefully measuring the distances between the horizontal bands on the Stopping Times Graph. As I went to the right, the distances seemed to be exponentially increasing, getting further and further apart. I decided to test this theory by graphing the same Collatz World but with the x-axis logarithmic...

... and was startled by the kind of "pleasant" regular looking graphs I got back. Let me show you what I mean. Start by taking the Stopping Times Graph, in particular, and examining it all the way out to n=100,000 instead of just n=10,000.

Now zoom in on the just the little rectangular area in the above image.

... and BAM!! What we see is a very regular, controlled pattern! What before was a super-weird unpredictable mess now reveals itself to be a very nice and periodic. It appear that the whole landscape is actually just one particular rectangular "tile" that keeps repeating. We've unlocked a very cool secret, hidden within the Stopping Times Graphs.

What's wild is that this "tiling" discovery seems to hold true in all Collatz Worlds! Let me show you a few examples...

The x3.25+7.5 World

The x3.564+1 World

The x2.9+11 World

And perhaps most importantly...

The x3+1 World

The finding of this fascinating "tiling" phenomenon was the first discovery (the tip of the iceberg, if you will) in a series of findings that would later go on to shed a great deal more insight into what the Stopping Times Graph actually is and why it operates the way it does. We will cover these issues a little later on in this chapter.