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: $A$

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.

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.