Polytropy

From the poster depicting a few von Neumann natural numbers, I created this animation. The moving image no longer depicts natural numbers in the sense of the poster, since there is no infinite descending chain of natural numbers. There is an infinite ascending chain of them; but the poster does not actually depict such a chain as nested circles. So running the animation in reverse would not give a correct suggestion of the original poster, even if it were of infinite size.

However, if the sense of membership were reversed, as suggested by Burak Kaya in a comment on the last post, then the animation shows us climbing through, say, 1, 2, 4, 5, 7, 8, and so on, skipping multiples of 3.

As I noted in a comment there, I don’t think this animation depicts a fractal in the fullest sense. You can find the whole image, not in all parts, but only in some.

I record here how easy it was to make the animation, once I understood how easy it was— and once I had the appropriate pdf file, created through LaTeX and the pstricks package. (It was possible to make LaTeX do most of the work for this.) From the file used for the original poster, I created a 64-page document, each showing the circles at a different magnification. I used pdf2jpg.net to convert the pdf file into 64 jpg files. In the GIMP Image Editor, using “Open as Layers”, I opened all of these jpg files (holding down the Control button to select all of them at once). Then I used “Save As…” and saved the whole collection as a gif file. This required allowing the program to convert them to an animation, which is what I wanted anyway.

Various websites have advice on how to create an animation with GIMP, but they seem to assume you are creating your little scene from scratch; so they introduce actions that were needless in the present case.