Dragon curve coding, part II

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:.1, 2, 3, 2, 3, 4, 3, 22, 1, 2, 3, 4, 3, 2, 33, 2, 1, 2, 3, 2, 3, 42, 3, 2, 1, 2, 3, 4, 33, 4, 3, 2, 1, 2, 3, 24, 3, 2, 3, 2, 1, 2, 33, 2, 3, 4, 3, 2, 1, 22, 3, 4, 3, 2, 3, 2, 1...OK, this is an important type of Gray Code map which in any Knight's move with wrap-arounds, there's one a "1" difference.In any row or column, there's a binomial frequency as to (1, 2, 3, 4) being (1, 3, 3, 1).Let's stop at this point and just say that for such 2^n x 2^n matrics, using the initial sequence S(n) = (1, 2, 3, 4, 5,....).the SUMS of terms in the matrices = (1, 6, 32, 160, 768,...) since in the 8x8 matrix each row has a sum = 20 and there's 8 such rows = 160.Before completing all of the connections, we can introduce the conclusion. We have created a type of 2^n x 2^n Gray Code Karnaugh map in which we can map terms in any sequence (S(n)) such that there will be a binomial frequency of terms in each row and column.Briefly, in your 1974 paper, you have the square of Pascal's triangle =Q. Just multiply that square * a diagonalized vector:V:10, 20, 0, 40, 0, 0, 80, 0, 0, 0, 16,...Getting Triangle A038208. Bring that up and you will find a "CYVIN" reference. I think he's retired and lives in Oslo. There's 2 of them - could be Father and Son.At any rate, any cites to "CYVIN" always indicate important applications to organic chemistry. So triangle A038208 = Q * V =12, 24, 8, 48, 24, 24, 8...which is symmetrical where as the square of Pacal's triangle is not symmetrical:12, 14, 4, 18, 12, 6, 1... = triangle T.Then triangle T * [1, 2, 3, 4, 5,...] = the sequence (1, 6, 32, 160, 768,...) showing the SUMS of terms in the 2^n x 2^n matrices.To conclude for now, we have introduced triangle T (not in your 1974 paper); which can be linked to a Gray Code map derived from the lengths of continued fractions in one half of the Infinite Farey Tree.Triangle T is symmetrical and has important applications to organic chemistry per citation in A038208 (Cyvin).Next, we will show further connections to your Harter-Heighway "Dragon Curve"....to be continued.