These images show the complex dynamics of some generalizations of the 3x+1 problem. They were originally created by Jeff Dumont and Cliff Reiter at Lafayette College to augment our paper "Visualizing Generalized 3x+1 Function Dynamics", Computers & Graphics, 25 5 (2001) 883-898. Some new additional material has been added. Click on the thumbnail to see the larger image or animation.

	The Escape Time
	The Stopping Time
	The Total Stopping Time
	The Coefficient Stopping Time for T(x)
	The Coefficient Stopping Time for T*(x)
	The Coefficient Stopping Time for T**(x)

	The Stopping Time
	The Escape Time
z=-17 z=-1 z=1 z=18	Zooms of the escape time
<	Annimation zoom into escape time of the winding generalization centered at z = 17.
	Cliff and Jeff with a hanging of images of the winding generalization created for APL 2001 (a mechanical break down prevented it from arriving, so wasn't shown there).

Here b=0 gives T(x), b=-0.5 gives W(x) and b runs from 0 to -1 in the aninmation (3.4M).

Animation of the Stopping Time for the ax+1 function changing from x+1 to 5x+1 (978k).

Winding Escape time zoom.

	The Escape Time
	The Stopping Time
	The Total Stopping Time

Here a runs from 1 to 5 for the animation (623k).

Preprints/Abstracts
Visualizing Generalized 3x+1 Function Dynamics [Preprint] [Abstract].

Real Dynamics of a 3-Power Extension of the 3x + 1 Function [Preprint] [Abstract].

Jsoftware Scripts
A J6.02 script 3x+1d.ijs that creates several images of this type. Original out of date script 3x+1.ijs
Note that J6.02 and graphics/fvj3, and media/image3 addons are required

Links

to Jeff Lagarias' 3x+1 page with papers and annotated bibliography.
to Eric Roosendaal's The 3x + 1 class record search
to Mike Keith's Music of hailstones
to Cliff's Gallery of Fractals, Chaos and Symmetry
to Cliff's Home Page