# 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. It is a fractal object similar in its construction to the dragon curve and the Hilbert curve.

 A fourth-stage Gosper curve The line from the red to the green point shows a single step of the Gosper curve construction.

## Algorithm

### 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.

### 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`.

## 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.