Space-filling
Gosper curve (flowsnake)
Bill Gosper’s curve, popularised by Martin Gardner in 1973, fills the so-called Gosper island, a hexagonal fractal region that tiles the plane. The pun “flowsnake” combines “snowflake” and “flow”.
At a glance
| Discoverer | Bill Gosper, c. 1973 |
|---|---|
| Hausdorff dimension | 2 (space-filling) |
| Self-similar copies | 7 copies at scale 1/√7 |
| L-system angle | 60° |
L-system
axiom: A
rules:
A → A-B--B+A++AA+B-
B → +A-BB--B-A++A+B
angle: 60°Gosper island
The region the curve fills, the Gosper island, is itself a fractal with hexagonal symmetry, and seven copies of the island tile a larger one of the same shape. It is one of the simpler rep-tiles that exist at fractal scales.
Why √7 instead of an integer?
The Gosper curve lives on the triangular (hexagonal) lattice. Each recursive step covers a hexagonally-oriented domain that grows by a factor of √7 in linear size and 7 in area: hence seven copies at scale 1/√7.
References
- Gardner, M., “Mathematical Games,” Scientific American, December 1976.
- Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman, 1982.
- Prusinkiewicz, P. & Lindenmayer, A., The Algorithmic Beauty of Plants, Springer, 1990.
- Hilbert curve · L-systems
Try it
Run an interactive playground at /tools/lsystems.