Plane Figure Fractals Discussion
Student: So fractals like
Sierpinski's Triangle and
Sierpinski's Carpet have recursion, because they each
have an initiator and a
generator. Is this what it takes to be a
Mentor: That's part of it. Do you remember what else we've discussed?
Student: Well, there is self-similarity too.
Mentor: Good. Here's something else to think about:
Student: These all seem to be contradictory statements.
Mentor: This is why infinity was such a hard concept to
understand for so long and there are still many debates about it.
Student: OK, I've seen lots of fractals now; what makes a fractal a fractal???
Mentor: Let's list the properties they all have in common:
- All were built by starting with an "initiator" and
"iterating" using a "generator." So we used recursion.
- Some aspect of the limiting object was infinite (length, perimeter,
surfacearea) -- Many of the objects got "crinklier."
- Some aspect of the limiting object stayed finite or 0 (area,
- At any iteration, a piece of the object is a scaled down,
otherwise identical copy of the previous iteration
Mentor: These are the characteristics that Benoit Mandelbrot
(who invented the term) ascribed to
Regular Fractals in 1975.