Gosper curve

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.

Algorithm

Lindenmayer System

The Gosper curve can be represented using an L-System with rules as follows:

• Angle: 60°
• Axiom: $A$
• Replacement rules:
  • $A \mapsto A-B--B+A++AA+B-$
  • $B \mapsto +A-BB--B-A++A+B$

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.

Logo

A Logo program to draw the Gosper curve using turtle graphics (online version):

to rg :st :ln
make "st :st - 1
make "ln :ln / sqrt 7
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 / sqrt 7
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.

Properties

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.

See also

• List of fractals by Hausdorff dimension

External links

• http://kilin.u-shizuoka-ken.ac.jp/museum/gosperex/343-024.pdf
• http://kilin.clas.kitasato-u.ac.jp/museum/gosperex/343-024.pdf
• http://www.mathcurve.com/fractals/gosper/gosper.shtml (in French)
• http://mathworld.wolfram.com/GosperIsland.html
• http://logo.twentygototen.org/mJjiNzK0
• http://80386.nl/projects/flowsnake/