The Gosper curve, also known as Peano-Gosper Curve, named after Bill Gosper, also known as the flowsnake (a spoonerism of snowflake), is a space-filling curve. It is a fractal object similar in its construction to the dragon curve and the Hilbert curve.
|A fourth-stage Gosper curve||The line from the red to the green point shows a single step of the Gosper curve construction.|
The Gosper curve can be represented using an L-System with rules as follows:
- Angle: 60°
- Replacement rules:
In this case both A and B mean to move forward, + means to turn left 60 degrees and - means to turn right 60 degrees - using a "turtle"-style program such as Logo.
to rg :st :ln make "st :st - 1 make "ln :ln / 2.6457 if :st > 0 [rg :st :ln rt 60 gl :st :ln rt 120 gl :st :ln lt 60 rg :st :ln lt 120 rg :st :ln rg :st :ln lt 60 gl :st :ln rt 60] if :st = 0 [fd :ln rt 60 fd :ln rt 120 fd :ln lt 60 fd :ln lt 120 fd :ln fd :ln lt 60 fd :ln rt 60] end to gl :st :ln make "st :st - 1 make "ln :ln / 2.6457 if :st > 0 [lt 60 rg :st :ln rt 60 gl :st :ln gl :st :ln rt 120 gl :st :ln rt 60 rg :st :ln lt 120 rg :st :ln lt 60 gl :st :ln] if :st = 0 [lt 60 fd :ln rt 60 fd :ln fd :ln rt 120 fd :ln rt 60 fd :ln lt 120 fd :ln lt 60 fd :ln] end
The program can be invoked, for example, with
rg 4 300, or alternatively
gl 4 300.
The constant 2.6457 in the program code is an approximation of √7.
The space filled by the curve is called the Gosper island. The first few iterations of it are shown below:
The Gosper Island can tile the plane. In fact, seven copies of the Gosper island can be joined together to form a shape that is similar, but scaled up by a factor of √7 in all dimensions. As can be seen from the diagram below, performing this operation with an intermediate iteration of the island leads to a scaled-up version of the next iteration. Repeating this process indefinitely produces a tessellation of the plane. The curve itself can likewise be extended to an infinite curve filling the whole plane.
- http://www.mathcurve.com/fractals/gosper/gosper.shtml (in French)