Cantor Dust

The Cantor set is a classical fractal generated by iteratively removing the middle third of a collection of line segments. To begin, start with a line segment from 0 to 1.

___________________________

Removing the middle third results in segments (0, 1/3) and (2/3, 1).

_________         _________

Removing the middle third of each of the remaining segments results in (0, 1/9), (2/9, 3/9), (6/9, 7/9), and (8/9, 1).

___   ___         ___   ___

And again:

_ _   _ _         _ _   _ _

The limit of this process is the Cantor set, which has Hausdorff dimension 0.6309.

Cantor set stages 0 through 6

Cantor dust is a two-dimensional fractal figure generated beginning with a square; with each iteration, remove the middle third horizontal and vertical stripe of each square in the figure. (Compare this process with the Sierpinski carpet process.)

The following figure shows the first few stages:

Cantor dust

The Hausdorff dimension is 1.2619.

The same process can be carried out in three dimensions:

Cantor dust in 3D

(Compare this process with the Menger sponge generation process.)

Designed and rendered using Mathematica 2.2 for the Apple Macintosh and (much, much later) 7.0 for Microsoft Windows.

Copyright © 1996–2017 Robert Dickau

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