Gosper curve

The Gosper curve, also known as Peano-Gosper Curve,[1] 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 object similar in its construction to the dragon curve and the Hilbert curve.

The Gosper curve can be used also to efficient hierarchical hexagonal clustering and indexing.[2]

Algorithm

Lindenmayer system

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

  • Angle: 60°
  • Axiom:
  • 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.

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 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

References

  1. Weisstein, Eric W. "Peano-Gosper Curve". MathWorld. Retrieved 31 October 2013.
  2. "Hierarchical Hexagonal Clustering and Indexing", 2019, https://doi.org/10.3390/sym11060731
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