A Moore curve (after E. H. Moore) is a continuous fractal space-filling curve which is a variant of the Hilbert curve. Precisely, it is the loop version of the Hilbert curve, and it may be thought as the union of four copies of the Hilbert curves combined in such a way to make the endpoints coincide.
Because the Moore curve is plane-filling, its Hausdorff dimension is 2.
The following figure shows the initial stages of the Moore curve.
Representation as Lindenmayer system
- Alphabet: L, R
- Constants: F, +, −
- Axiom: LFL+F+LFL
- Production rules:
- L → −RF+LFL+FR−
- R → +LF−RFR−FL+
Here, F means "draw forward", + means "turn left 90°", and − means "turn right 90°" (see turtle graphics).
Like the Hilbert curve, the Moore curve can be extended to three dimensions:
- A. Bogomolny, Plane Filling Curves from Interactive Mathematics Miscellany and Puzzles, Accessed 7 May 2008.