Gosper curve
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
- List of fractals by Hausdorff dimension
- M.C. Escher
References
External links
- NEW GOSPER SPACE FILLING CURVES
- FRACTAL DE GOSPER (in French)
- Gosper Island at Wolfram MathWorld
- Flowsnake by R. William Gosper
- v
- t
- e
- Fractal dimensions
- Assouad
- Box-counting
- Higuchi
- Correlation
- Hausdorff
- Packing
- Topological
- Recursion
- Self-similarity
system
- Barnsley fern
- Cantor set
- Koch snowflake
- Menger sponge
- Sierpinski carpet
- Sierpinski triangle
- Apollonian gasket
- Fibonacci word
- Space-filling curve
- String
- T-square
- n-flake
- Vicsek fractal
- Hexaflake
- Gosper curve
- Pythagoras tree
- Weierstrass function
fractals
- "How Long Is the Coast of Britain?"
- Fractal art
- List of fractals by Hausdorff dimension
- The Fractal Geometry of Nature (1982 book)
- The Beauty of Fractals (1986 book)
- Chaos: Making a New Science (1987 book)
- Kaleidoscope
- Chaos theory