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The Gosper curve has a sort of dual flowsnake which is a space-filling curve on a hex lattice. The unit motif traces a 7 hex 'super-hexagon' comprised of the origin and its six neighbors in the three +/- axis directions, which get connected into a super-super-hex of 49 hexes and so on.

I know there are plenty of space-filling curves on a Cartesian (square) lattice like the Hilbert curve, z-order curve, Grey-coded curve etc. But the flowsnake construction makes me wonder if there's an analogue whose basic unit links the origin with its four axis neighbors $(x,y) = (0, 0), (\pm 1, 0), (0, \pm 1)$ into a + shape. The recursive packing of these into a larger rotated version seems to work in a similar way (bad ascii art below), but I'm not sure how the path would trace.

Anyone run across a space filling curve that looks like this?

        . . 4 . . . .
        . 4 4 4 . 1 .
        . . 4 0 1 1 1 
        . 3 0 0 0 1 .
        3 3 3 0 2 . .
        . 3 . 2 2 2 .
        . . . . 2 . .

Illustration of flowsnake construction connecting seven instances of the red 7 hexagon motif to create a self-similar 49 hex motif sketched in green, in turn connected to form a 777 hex motif, and so on

patricksurry
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  • rather than describing the proposed 5 square unit on the square lattice as a + sign, i guess calling it a rotated 'super-square' would be more apt to extend the analogy – patricksurry Feb 14 '23 at 23:18

1 Answers1

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Kevin Ryde, author of the planepath package with many interesting 2D curves, pointed me at Mandelbrot's quartet.

Like the Gosper curve, we can construct a dual that fills the cartesian lattice based on a 5 unit 'super-cell' that recursively tiles the square grid.enter image description here

patricksurry
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