David Makin's Fractal Blog
The Peano Curve:
Replace the middle third with a figure of 8:
Now repeat, replacing the middle third of each line segment with a figure of 8 rotated according to the orientation:
If you were to continue the construction it results in a solid diamond - a 2 dimensional object.
From a topolgical point of view it's important to note that the entire construction can be created without any crossings or double-backs ie. you could draw the whole thing starting from the left end and continue to the right end without removing pen from paper or doubling back or once crossing the line (touching but not crossing) eg. for the first stage it's:
How long would the line drawn be?
Is it really a 2 dimensional object?
This type of fractal curve (producing a 2 dimensional result) is known as a "space filling curve" and there are many of them.
Copyright © 2007 Ian Lewis & David Makin