Dragon curve coding, part I

. An "L" system for fractal curves is the series of bits that encodes the directions of moves, through which we can recover the entire fractal. First, we will show the connections between the binomial frequency, infinite farey tree continued fraction lengths, and the Dragon curve, then go on from there in next e mail. The "L" system for the Dragon Curve (Cf. the Mathworld entry on "Dragon Curve" - they have it shown right there. Its entryA014577 in OEIS: http://www.tinyurl.com/4zq4q shown as: (1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0,...). b. now, bring up A088696, lengths of continued fraction representations of infinite farey tree terms. (one half of the tree). This is a very important set of terms since your papers touch upon this theme repeatedly. The fractions are:1/21/3, 2/31/4, 2/5, 3/5, 3/41/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5...and the corresponding continued fraction representations are: [2][3), [1,2][4] [2,2], [1,1,2], [1,3][5], [3,2], [2,1,2], [2,3], [1,1,3], [1,1,1,2], [1,2,2], [1,4]...now simply write down the number of terms in each CF representation = 11, 21, 2, 3, 21, 2, 3, 2, 3, 4, 3, 2Now taking this, write down a R for leftmost term = Right = 1 in the Dragon curve if the NEXT term is greater. If less, write down an 0. This gets us the L system for the Dragon curve: (inserting an initial 0) we obtain rows which tend to the Dragon Curve L system mentioned in Mathworld:(0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0,...). ...note here that this is a binomial frequency, since in the 8 bit row we have one 1, three 2's, three 3's, and one 4.Similarly, the next row would have a binomial frequency of (1, 4, 6, 4, 1), and so on.OK, the foregoing are linear maps. We can create a 2-D map using the same terms as follows: Top row and left column we place(1, 2, 3, 2, 3, 4, 3, 2), with (1,1,1,1,....) on the diagaonal. If leftmost col = 1, then odd rows circulate from position (n,n) DOWN, while evens circulate UP from position (n,n). This is tricky but crucial. This is a new type of Gray Code MAP:.