Hilbert Curve

Hilbert cube

Dragon curve

Without spacing between, using it to choose colours

Watch it draw

Watch it draw again, this one looks more like a Hilbert curve b/c I spaced the vertices.

Make it tweetable!

https://twitter.com/josh_cheek/status/667502337812860929

ruby -e's=?F;3.times{s.gsub!?F,"F3F1F1F3F"};puts"\e[30H\e[47m#{s.gsub(/./){$.+=$&.to_i;"\e[#{%w[2C B 2D A][$.%4]}  \e[2D"*2if$&[?F]}}\e[m"'

https://twitter.com/josh_cheek/status/760532928803672064

ruby -e 'r=1..4;3.times{r=r.flat_map{|n|[0,0,1,2,3,0,0].map{|m|n+m}}};$><<r.map{|n|"\e[42m  \e[2D\e[#{n%2*2}#{"ACBD"[n%4]}"*2}*""<<"\e[59H"'

32 Segment curve

Square curve

Resources

• Paper on 3d Hilbert curves http://arxiv.org/pdf/1109.2323v1.pdf

• Representing a 3d Hilbert curve as an Lsystem http://math.stackexchange.com/questions/123642/representing-a-3d-hilbert-curve-as-an-l-system This is the definition I actually used

  lsystem Hilbert3D {
  
      set iterations = 3;
      set symbols axiom = X;
  
      interpret F as DrawForward(10);
      interpret + as Yaw(90);
      interpret - as Yaw(-90);
      interpret ^ as Pitch(90);
      interpret & as Pitch(-90);
      interpret > as Roll(90);
      interpret < as Roll(-90);
      rewrite X to ^ < X F ^ < X F X - F ^ > > X F X & F + > > X F X - F > X - >;
  }

• Hilbert curves you can draw https://bentrubewriter.wordpress.com/2012/04/26/fractals-you-can-draw-the-hilbert-curve-or-what-the-labyrinth-really-looked-like/ Provides Lsystems for a number of different 2d curves:

  • Hilbert
    • Axiom: A
    • A -> - B F + A F A + F B -
    • B -> + A F - B F B - F A +
  • Sierpinski Triangle
    • Axiom A
    • A -> B-A-B
    • B -> A+B+A
    • In this case we rotate 60 degrees with every turn, and A and B are both used to mean draw a line forward
  • Koch Curve
    • Axiom F++F++F
    • F -> F-F++F-F
    • Use 60 degree turns
  • Dragon Curve L-System:
    • Axiom: FX
    • X -> X+YF
    • Y -> FX-Y
    • Use 90 degree turns
  • Quadratic Fractal:
    • Axiom: F+F+F+F
    • F -> F+F-F
    • Use 90 degree turns
  • Koch Curve Variant:
    • Axiom = F
    • F -> F+F-F-F+F
    • Use 90 degree turns
  • Fractal Plant:
    • Axiom: X
    • X -> F-[[X]+X]+F[+FX]-X
    • F -> FF
    • Use 25 degree turns. When you encounter a ‘[‘ save the current angle and position and restore when you see ‘]’. This is an example of a recursive L-System.

• A bunch of 2d curves, plus some code for Hilbert in Mathematmica http://mathforum.org/advanced/robertd/lsys2d.html

  • Koch curve (F -> F+F--F+F, 60°):
  • 32-segment curve (F -> -F+F-F-F+F+FF-F+F+FF+F-F-FF+FF-FF+F+F-FF-F-F+FF-F-F+F+F-F+)
  • Hilbert curve (L -> +RF-LFL-FR+, R -> -LF+RFR+FL-)
  • Arrowhead curve (X -> YF+XF+Y, Y -> XF-YF-X, 60°)
  • Dragon curve (X -> X+YF+, Y -> -FX-Y)
  • Hilbert curve II (X -> XFYFX+F+YFXFY-F-XFYFX, Y -> YFXFY-F-XFYFX+F+YFXFY)
  • Peano-Gosper curve (X -> X+YF++YF-FX--FXFX-YF+, Y -> -FX+YFYF++YF+FX--FX-Y, 60°)
  • Peano curve (F -> F+F-F-F-F+F+F+F-F)
  • Quadratic Koch island (F -> F-F+F+FFF-F-F+F)
  • Square curve (X -> XF-F+F-XF+F+XF-F+F-X)
  • Sierpinski triangle (F -> FF, X -> --FXF++FXF++FXF--, 60°)

• The algorithmic beauty of plants http://algorithmicbotany.org/papers/#abop

• Someone's 3d Hilbert curve for Processing http://martinpblogformasswritingproject.blogspot.com/2011/06/3d-hilbert-fractal-in-pyprocessing.html

• Someone's 3d Hilbert curve for Mathematica http://robertdickau.com/lsys3d.html

• Some nice 2d Hilbert curve results https://lustrebox.wordpress.com/2012/06/17/fractals/

License

Just do what the fuck you want to