A question related to Joerg's: Assign coordinates (x,y,z) to the hexagonal plane tiling by projecting from the lattice section x+y+z = 0 in 3-space: for example, the central cell gets [0,0,0], and its neighbours [+2,-1,-1], [+1,-2,+1], [-1,-1,+2], [-2,+1,+1], [-1,+2,-1], [+1,+1,-2]. What function of x,y,z characterises the black cells in (extended indefinitely) RWG's hex flowsnake at http://gosper.org/hexflo.png ? WFL On 7/31/11, Joerg Arndt <arndt@jjj.de> wrote:
* Bill Gosper <billgosper@gmail.com> [Jul 31. 2011 19:00]:
(to go with the Penrose toilet paper) Neil, Corey, and I (over two sessions) worked out how, with tiles of two solid colors, to map a fill of the flowsnake (Gosper) curve onto a hex grid,<http://gosper.org/hexflo.png>
Nice!
a staggered square grid <http://gosper.org/staggerflo.png>, and a square grid <http://gosper.org/gridflo.png> wherein squares are considered joined at their NE and SW corners but not the other two. The illustrations are of three flowsnakes joined in a triangle to create a closed curve.
A few weeks(?) ago I had a burning desire to find this hex version in conjunction with either a math-fun thread or a project with the kids, but da*ned if I can remember what it was.
I recently spammed some short routines, one of them would produce what is an p.94 of the fxtbook:
bool bit_dragon3_turn(ulong &x) [...]
For the picture one has to replace each move by 120 degs by two moves by 60 degs.
The picture you made certainly reminded me of it.
Btw. would you know a "radix-something counting" method of your curve (similar to the routines I posted)?
Maybe it was just to emphasize to Erez Lieberman that hex grids and square grids aren't all that different for spacefilling purposes. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
cheers, jj
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