Gosper 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, also known as Peano-Gosper Curve, named after Bill Gosper, also known as the flowsnake (a spoonerism of snowflake), is a space-filling curve whose limit set is rep-7. It is a fractal curve similar in its construction to the dragon curve and the Hilbert curve.

The Gosper curve can also be used for efficient hierarchical hexagonal clustering and indexing.

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.

Python

A Python program, that uses the aforementioned L-System rules, to draw the Gosper curve using turtle graphics (online version):

import turtle

def gosper_curve(order: int, size: int, is_A: bool = True) -> None:
    """Draw the Gosper curve."""
    if order == 0:
        turtle.forward(size)
        return
    for op in "A-B--B+A++AA+B-" if is_A else "+A-BB--B-A++A+B":
        gosper_op_map[op](order - 1, size)

gosper_op_map = {
    "A": lambda o, size: gosper_curve(o, size, True),
    "B": lambda o, size: gosper_curve(o, size, False),
    "-": lambda o, size: turtle.right(60),
    "+": lambda o, size: turtle.left(60),
}
size = 10
order = 3
gosper_curve(order, size)

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 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
• M.C. Escher

External links

• NEW GOSPER SPACE FILLING CURVES
• FRACTAL DE GOSPER (in French)
• Gosper Island at Wolfram MathWorld
• Flowsnake by R. William Gosper