On 2019-08-28 09:34, Brad Klee wrote:

The article should also say something about limit-periodicity. For example try:

SetDragon[Root_, BinaryBranching_] := Append[ Flatten[Reap[FoldList[(Sow[ {x_Integer /; IntegerQ[ Divide[x - #1 - (2^#2[[2]])*BitXor[0, #2[[1]]], 2^(#2[[2]] + 1)]] :> 1, x_Integer /; IntegerQ[ Divide[x - #1 - (2^#2[[2]])*BitXor[1, #2[[1]]], 2^(#2[[2]] + 1)]] :> 0}]; #1 + (2^#2[[2]])* BitXor[0, #2[[1]]] + 2^(#2[[2]] - 1)) &, Root, Transpose[{BinaryBranching, Range[Length@BinaryBranching]}]] ][[2]]], _Integer :> "?"]

ArrayPlot[ Map[Range[0, 2^6] /. SetDragon[0, #] &, NestList[Append[#, RandomInteger[{0, 1}]] &, RandomInteger[{0, 1}, 1], 6]], Mesh -> True, ImageSize -> 500]

Wow, for the first time on Planet Earth, an actual brillathong keyboard. Brad, this is too dense for me. For starters, is a Fourier transform pure point the same as a periodic point? I can see why the Wikipedist dodged this can of worms. The simplest Fourier transform I can find for Heighway's Dragon has for its coefficients infinite products of non-triangular 3⨉3 matrices. first few dragon harmonics <http://gosper.org/FDrags128.gif> —rwg

This definition clearly shows relation of the tiling space to a binary tree, and also explains why the Fourier transform should have pure points. It also helps to explain why the dragon sequence shows up as a factor in Socolar-Taylor tiling, and in other hexagonal or square quadtree tilings.

--Brad

On Wed, Aug 28, 2019 at 10:30 AM Bill Gosper <billgosper@gmail.com> wrote:

...

is the usual mix of good and bad. Good: Illustration that 4 Dragons joined at the nose <

https://en.wikipedia.org/wiki/Dragon_curve#/media/File:Dragon_tiling1.svg>

...

fill a tile different from four joined at the tail < https://en.wikipedia.org/wiki/Dragon_curve#/media/File:Dragon_tiling2.svg>. Oops: Fails to note that twin1 < https://en.wikipedia.org/wiki/Dragon_curve#/media/File:Dragon_tiling3.svg> and twin2 < https://en.wikipedia.org/wiki/Dragon_curve#/media/File:Dragon_tiling5.svg> are *both* twindragons*.* (Svgs, but painted with a horrible fat brush that *ruins* the symmetry.) Howler: (Re: the bounding box) "Note that the dimensions 1, and 1.5 are limits <https://en.wikipedia.org/wiki/Limit_(mathematics)> and not actual values." There remains almost universal ignorance of Dragon curves being merely lousy plots of the (almighty) Dragon Function. Axis-aligned tangents to the Dragon image intersect it in (uncountable) Cantor sets! E.g., some points on the bottom edge (Im(z) = -1/3) have spacings {1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, . . .}/1023.

Next term: 4⨉171 - 1 = 683

...

This is *a276391**,* the spacings of *preimages* of quadruple points of the *Hilbert *curve! More incredibly, another subset of the tangent points along x-i/3 has differences {1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 170, . . .}/4080, with 170 instead of 171!

Next terms: 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 680, . . .

...

—rwg