Mathematica MathWorld Wolfram Science More »

HOME

TOPICS

LATEST

Hilbert and Moore 3D Fractal Curves

The Hilbert curve and the Moore curve are two famous plane-filling curves that can be extended to 3D space-filling curves. They have similar recursive constructions, here using L-systems. The curves shown in this Demonstration map points in three-dimensional space to points on a one-dimensional line, and thus have properties that make them useful for certain types of data manipulation, such as image processing.

Contributed by: Robert Dickau

Show Initialization Code

(20 lines omitted)

Snapshot 1: viewing multiple iterations at once and rotating the resulting image can assist in making the recursive construction of the curves clearer

Snapshot 2: one way in which the Hilbert curve and Moore curve differ is in the relative positions of the start and end points of the curve. While the Hilbert curve begins and ends in adjacent corners of the bounding cube, the Moore curve begins and ends in adjacent points in the resulting three-dimensional grid.

Snapshot 3: in this Demonstration, the Hilbert curve and the Moore curve begin with the same initial shape; it can be illuminating to switch between the Hilbert and Moore figures for a given iteration level

The Lebesgue curve has similar properties; while it is easier to compute, the Hilbert curve is often desirable in that no two consecutive points are very far apart.

Hilbert Curve (Wolfram MathWorld)

Plane-Filling Function (Wolfram MathWorld)

Hilbert and Moore Fractal Curves (The Wolfram Demonstrations Project)

Lebesgue 3D Curves (The Wolfram Demonstrations Project)

"Hilbert and Moore 3D Fractal Curves" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/HilbertAndMoore3DFractalCurves/

Contributed by: Robert Dickau

Report an issue »

Interact with Demonstrations using the latest version of the free Mathematica Player—Download Now

Browse all topics

	Source page:
	Email	Name (optional)
	Occupation (optional)	Organization (optional)
	I use Mathematica often occasionally never
	Country (optional) Remember me
	Note: Please do not include anything you consider confidential or proprietary. Your message and contact information may be shared with the author of any specific Demonstration for which you give feedback, but will not otherwise be published or distributed. Privacy Policy »

Wolfram Research

Site Index

RSS

Atom