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