Gosper curve
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Related on Lably
Through Tessellation, Plane (mathematics) and Similarity (geometry)
The Gosper curve can also be used for efficient hierarchical hexagonal clustering and indexing.
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.
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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.