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Space-filling curve From Wikipedia, the free encyclopedia
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]
The Gosper curve can be represented using an L-system with rules as follows:
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.
The space filled by the curve is called the Gosper island. The first few iterations of it are shown below:
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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.
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