A Moore curve (after
E. H. Moore) is a
continuous fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
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
The Moore curve can be expressed by a
rewrite system (
L-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).
Generalization to higher dimensions
There is an elegant generalization of the
Hilbert curve to arbitrary higher dimensions. Traversing the polyhedron vertices of an n-dimensional hypercube in
Gray code
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).
For example, the representati ...
order produces a generator for the n-dimensional Hilbert curve. Se
MathWorld
To construct the order N Moore curve in K dimensions, you place 2
K copies of the order N−1 K-dimensional Hilbert curve at each corner of a K-dimensional hypercube, rotate them and connect them by line segments. The added line segments follow the path of an order 1 Hilbert curve. This construction even works for the order 1 Moore curve if you define the order 0 Hilbert curve to be a geometric point. It then follows that an order 1 Moore curve is the same as an order 1 Hilbert curve.
To construct the order N Moore curve in three dimensions, you place 8 copies of the order N−1 3D Hilbert curve at the corners of a cube, rotate them and connect them by line segments. This is illustrated by
Wolfram Demonstration
See also
*
Hilbert curve
*
Sierpiński curve
*
z-order (curve)
*
List of fractals by Hausdorff dimension
References
* Moore E.H. On certain crinkly curves.– Trans. Amer. Math. Soc. 1900, N1, pp. 72–90.
External links
*
A. Bogomolny''Plane Filling Curves from Interactive Mathematics Miscellany and Puzzles'' Accessed 7 May 2008.
{{Fractals
Fractal curves