Colonel 'Sierpinski' Korn

Prodding the envelope by disallowing "backward" moves i.e. the relentless march of progress must be to the right, an old familiar Friend predictably emerges.

I pushed it to 40 iterations, just for amusements sake, but it was clear after 18 that we were entering a Sierpinski Gasket, and once in, you never come out.

[EDIT] That is obviously not a Sierpinski, I don't know what I was thinking. It is familiar though, and I think I have seen it somewhere in Wolfram's Big Book of Diagrams.*

[EDIT] After scanning Wolfram's oeuvre, I'd like to say Rule 110, except its not. I think it was just the semi-regular triangles that are throwing me off,

By disallowing "backward" moves, we disallow iteration on anything except the previous generations output.
Unfortunately, the interesting parts of the Dho-Na curve are the parts on the -x side which is unreachable this way. I only calculated the (+x,+y) section, as the -x quadrants are unreachable, and (+x,-y) is just a flopped image.
The next step is to apply Thue-Morse to the tree algorithm, I think.
I was entertained by this paper on Koch, Thue Morse and Turtles, an almost Carrolean combination.

[EDIT] This is why you don't draw this stuff by hand - there a couple of typos in the above gasket.
Holes in the Gasket!

...For a moment I wanted the title to link to the Colonel Korn reference, rather than to the blog post, which suggests the need for some sort of pointers-to-pointers in hypertext. But than I came to my senses.

*'A new kind of science', Wolfram (2002), also known as the Liber Glyphic Iteraticus.