Skipping Iterations
More lines

We can skip Ulam iterations when layering them to bring out more patterns.

By starting at the second iteration and then rendering every second iteration after that we are looking at a layering of the 2nd, 4th, 6th etc. Ulam iteration.

After reaching 150 iterations (75 iterations rendered... we are skipping every second one) we see the following:


What really shows up in this one is another dimension of Exclusion lines that have not been seen before.

The exclusion lines are hard to see because the Exclusion lines that we already know about are much more prominent so your eyes are drawn to them... so, here is an image that highlights one of poles and the exclusion lines that radiate from it, I have drawn the lines very roughly! (and here I was thinking I would get off without any more line drawing)


Once you know where they are they are reasonably easy to see on the untouched image.

The next axis is a tough one... the exclusion lines that radiate from the pole are very hard to see... this one is stretching the limits a little bit I think... but they are clearer to me because I can look at all the layers around this 150th. Anyway, below are are the exclusion lines that I think radiate from the other new pole.




Starting with Ulam iteration 4, and skipping 3 in between we are looking at layers of Ulams 4, 8, 12, 16 etc...

After reaching 500 iterations (125 iterations rendered... we are rendering every 4th) we see the following:



There is another plane of Exclusion lines in there...




Below is a 600 x 600 pixel image of 360 layered Ulam iterations, rendered without skipping any iterations. The image started out 3000 pixels wide... that makes 9 million integers being rendered on the first Ulam iteration. The image was resized to 600 x 600 pixels because it is just too large to look at 3000 x 3000 pixels (I wish I had a bigger screen!)


At this scale there are other Axis visible, some are not that clear in the above image because of the degredation when saving a compressed version of the image.



Copyright © 2007 - H Rudd