## Chapter 5.6: The Relationship Between the Stopping Time Graph and the Tree Diagram

So if we think all the way back to the beginning of this research project, my original obsession was with these three pictures, the three classical ways to visually represent the x3+1 world:

While I've mostly been spending my nights and days thinking about the two images on the left (especially the Stopping Times Graph), other mathematicians have taken an entirely different approach to the problem and have gotten lost out in the maze of the Tree Diagram to the right. I never really thought my research would contact theirs... until something absolutely wild happened!

I had the following epiphany one night...

Take the standard x3+1 stopping times image:

Logarithm the x-axis and zoom in on the bottom:

You get this image:

If we draw all the connecting arrows, we get:

And now we perform the following wild transformation!...

Revealing that the Stopping Times Graph actually was the Tree Diagram in disguise all along!

Can you see it now?

This cool discovery has played a key role in almost all of my research that has followed thereafter.