Moore curve

# Moore curve

Discussion

Encyclopedia
A Moore curve is a continuous fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

space-filling
Space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square...

curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

which is a variant of the Hilbert curve
Hilbert curve
A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling curves discovered by Giuseppe Peano in 1890....

. 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
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...

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
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. What is considered are rewriting systems...

(L-system
L-system
An L-system or Lindenmayer system is a parallel rewriting system, namely a variant of a formal grammar, most famously used to model the growth processes of plant development, but also able to model the morphology of a variety of organisms...

).
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
Turtle graphics
Turtle graphics is a term in computer graphics for a method of programming vector graphics using a relative cursor upon a Cartesian plane...

).

Like the Hilbert curve, the Moore curve can be extended to three dimensions:

• A. Bogomolny, Plane Filling Curves from Interactive Mathematics Miscellany and Puzzles

http://www.cut-the-knot.org/do_you_know/hilbert.shtml, Accessed 07 May 2008.

• Hilbert curve
Hilbert curve
A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling curves discovered by Giuseppe Peano in 1890....

• Sierpiński curve
Sierpinski curve
Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \rightarrow \infty completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling...

• z-order (curve)
Z-order (curve)
In mathematical analysis and computer science, Z-order, Morton order, or Morton code is a space-filling curve which maps multidimensional data to one dimension while preserving locality of the data points. It was introduced in 1966 by G. M. Morton...

• List of fractals by Hausdorff dimension