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, named after Bill Gosper, also known as the Peano-Gosper Curve[1] and 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.[2]

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.

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

References

  1. ↑ Weisstein, Eric W. "Peano-Gosper Curve". MathWorld. Retrieved 31 October 2013.
  2. ↑ Uher, Vojtěch; Gajdoš, Petr; Snášel, Václav; Lai, Yu-Chi; Radecký, Michal (28 May 2019). "Hierarchical Hexagonal Clustering and Indexing". Symmetry. 11 (6): 731. doi:10.3390/sym11060731. hdl:10084/138899.
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