* Fred Lunnon <fred.lunnon@gmail.com> [Nov 07. 2016 17:26]:
OK, I get it now. Though what continues to baffle me is how you came to be looking for such an object in the first place!
Answer below your message (SPOILER).
The version below makes the symmetry more apparent when viewed without proportional spacing (view source, or drag/drop to text file).
Thanks for that one, hope it encourages other to look harder!
WFL
[...]
It is a tile: the thing tiles the plane as a square does. Any such (4-symmetric) tile corresponds to plane-filling curves on the square grid (this special tile corresponds to just one curve). The search space for curves of order 257 is about 10^70, so way out for direct search (as done before). Looking for the tiles does speed up the search immensely (e.g., by the factor of about 10^7 for not-too-big orders). Now I thought "it's just backtracking, that cannot be hard". It is backtracking, but in 2 dimensions. So I did not yet find an efficient way to obtain all tiles (an avoid duplication). This one is in the class of tiles I currently can find: there is a purely linear way from the center to the out-most points. I computed all those tiles for all orders (and their decompositions into two squares) up to 300. They look (IMO) stunning. Drawback: smaller orders tend to have visual elements of swastikas in them, so I picked one that does NOT look that way. Best regards, jj P.S.: for a numeration system with base (15 + 8*i) take as digits the 257 marked squares, then the generalization of the unit square looks (about) like shown in the image. P.P.S: Is backtracking in 2D really nontrivial, or am I sitting on my brain? I need to find all 4-symmetric edge-to-edge connected arrangements for orders n = 4*k + 1. (Adding the tiling condition is trivial). P.P.P.S: Here is another one, for order 281 = 16^2 + 5^2. As before, the x and y are the shifts for the four adjacent tiles, take (+x, +y), (+y, -x), (-x, -y), and (-y, +x).
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