A Moore curve (after
E. H. Moore
Eliakim Hastings Moore (; January 26, 1862 – December 30, 1932), usually cited as E. H. Moore or E. Hastings Moore, was an American mathematician.
Life
Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, di ...
) is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
fractal space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...
which is a variant of the
Hilbert curve
The Hilbert curve (also known as the Hilbert space-filling 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 Peano curves discovered by Giuseppe ...
. Precisely, it is the
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
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
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
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
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduc ...
(
L-system
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into som ...
).
: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
In computer graphics, turtle graphics are vector graphics using a relative cursor (the "turtle") upon a Cartesian plane (x and y axis). Turtle graphics is a key feature of the Logo programming language.
Overview
The turtle has three attribut ...
).
Generalization to higher dimensions
There is an elegant generalization of the
Hilbert curve
The Hilbert curve (also known as the Hilbert space-filling 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 Peano curves discovered by Giuseppe ...
to arbitrary higher dimensions. Traversing the polyhedron vertices of an n-dimensional hypercube in
Gray code 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
The Hilbert curve (also known as the Hilbert space-filling 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 Peano curves discovered by Giuseppe ...
*
Sierpiński curve
Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \to \infty completely fill the unit square: thus their limit curve, also called the Sierpiń ...
*
z-order (curve)
In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It i ...
*
List of fractals by Hausdorff dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illus ...
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