[math-fun] flowsnake bathroom tile
(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> 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. Maybe it was just to emphasize to Erez Lieberman that hex grids and square grids aren't all that different for spacefilling purposes. --rwg
* 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
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
* Joerg Arndt <arndt@jjj.de> [Aug 01. 2011 10:29]:
[...]
I can obtain Gosper's hex curve via first using the 'usual' L-system Start: X Rules: X --> X+Y++Y-X--XX-Y+ Y --> -X+YY++Y+X--X-Y + --> + - --> - ------------- 0: (#=1) X 1: (#=15) X+Y++Y-X--XX-Y+ 2: (#=113) X+Y++Y-X--XX-Y++-X+YY++Y+X--X-Y++-X+YY++Y+X--X-Y- ... followed by a final substitution X --> X-X+ Y --> -X+X (This comes form looking at the construction and going to the centers of the hexagons). It gives: X-X++-Y+Y++-Y+Y-X-X+--X-X+X-X+--Y+Y++-X-X++-Y+Y-Y+Y++-Y+ ... Now we remove all "+-" to get X-X+Y+Y+Y+Y-X-X-X-X+X-X-Y+Y+X-X+Y+Y-Y+Y+Y+ ... Now we look at just the turns (here I take the complement from the above): +----++++-++--+--+----++++--+--+----++++-++-- ... (Not in the OEIS, apparently nothing about Gospers curve btw.). See http://www.jjj.de/frac/ for a rendering (file hex-gosper.pdf ) and file hex-gosper-closed.pdf for a closed curve.
Btw. would you know a "radix-something counting" method of your curve (similar to the routines I posted)?
There ought to be a radix-14 algorithm. Splitting into chunks of 14 we get: 01234567890123 +----++++-++-- +--+----++++-- +--+----++++-+ +----++++-++-+ +----++++-++-- +----++++-++-+ +--+----++++-- +----++++-++-- +--+----++++-+ +--+----++++-- +--+----++++-- +----++++-++-+ ... So modulo 14 some turns are fixed. The remaining come in two groups: 3 and 9 are one thing and 5,6,7 its complement, while track 13 is something else.
[...]
Still, no success with finding some neat algorithm (cannot even get a single L-system without some final substitution). cheers, jj
participants (3)
-
Bill Gosper -
Fred lunnon -
Joerg Arndt