I have a question on combinatorial fractal geometry.

I drew the dragon curve on a piece of graph paper. For this particular figure the total distance of the sides increases by a factor of squareroot(2) each iteration, and the number of full boxes at each level is: 1, 4, 11, 28, 67, 152, 335, 724, 1539, 3232, 6727, 13900, 28555...   Here are two resources on this fractal:



as you follow the curve, at no point did a line create two boxes at the same time. Rather, it was always possible to tell which box would be created and numbered next, even when the figure ran up against itself and started creating boxes rapidly. 
The mean of all the numbers for a given translation of the dragon figure increased in a progression that corresponded to what I think is the equation of the curve: 1, 3, 8, 20, 47, 107, 238...  a_n = a_(n-1) + a_(n-2) + a_(n-3) + 2^(n-3).    It seems to me that some of these numbers might correspond in other interesting ways. For example, 1 could fold into 4, then 7, then 20. I’m imagining numbering the boxes by drawing the fractal using the lindenMayer system and then folding it to see if there’s an interesting combination of numbers. I tried checking in the online encyclopedia of integer sequences oeis but I didn’t find anything.

My basic question is - is numbering the boxes in that way useful? I haven’t seen any resources with these boxes numbered like this

enter image description here

  • $\begingroup$ Useful for what? $\endgroup$ Nov 13 '20 at 5:16
  • $\begingroup$ Ok if not useful, interesting ? $\endgroup$
    – hachiko
    Nov 13 '20 at 5:18
  • $\begingroup$ Interesting how? Or... how as interesting as a million other questions? $\endgroup$ Nov 13 '20 at 5:54
  • $\begingroup$ Maybe there’s an integer sequence here that has relevance in another place. I’m not sure. I spend most of my time on stackoverflow and this is my first time venturing into the math side of things. It’s not my background. I understand I’m not formulating a question with a clear set of guidelines for an answer but I tried to document things as clearly as I could. $\endgroup$
    – hachiko
    Nov 13 '20 at 14:19
  • $\begingroup$ A posting should be better motivated than "here's a problem... solve it for me." $\endgroup$ Nov 13 '20 at 16:26

"My basic question is - is numbering the boxes in that way useful?"

No. I see no use for this numbering. It is arbitrary and gives no insight, especially because whatever question that might be related is unclear and unmotivated.

  • $\begingroup$ I’m trust your answer but why would the ulam spiral, for example, which is a sequence of numbers in a counter-clockwise square, be interesting (primes related to quadratic formulas) but you’re sure that this ordering of numbers in a different geometry (fractal) is arbitrary ? $\endgroup$
    – hachiko
    Nov 13 '20 at 17:39
  • $\begingroup$ @hachiko: This should answer your last comment: youtube.com/watch?v=EK32jo7i5LQ $\endgroup$ Nov 13 '20 at 17:43

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