* Joerg Arndt <arndt@jjj.de> [Aug 01. 2011 10:29]:
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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.
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Still, no success with finding some neat algorithm (cannot even get a single L-system without some final substitution). cheers, jj