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 . .
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.