It's interesting that the Collatz tapes look like wavy sheets, like a photo of a curtain with a shadow cast over one half. I wonder if there is such a higher-dimensional shape that's underlying these 2D plots, and what that shape might look like from different points of view.

I think a large part of that is the read head wandering left and right. For any machine that halts in a nontrivial number of steps it would be expected that the head doesn't move monotonically.