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 fractal?

Mentor: That's part of it. What about the other stuff we talked about?

Student: Well, there is self-similarity too.

Mentor: Good. Here's something else to think about:

• In the Hilbert Curve an infinite curve filled a finite amount of space.
• In the Koch Snowflake an infinite border contained finite space.
• In the Sierpinski Triangle and Sierpinski Carpet the area of the final figure was 0 - yet we could still see it.

Student: These all seem to be contradictory.

Mentor: This is why infinity was such a hard concept to pin down for so long.

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, surface area) -- Many of the objects got "crinklier."
• Some aspect of the limiting object stayed finite or 0 (area, volume, etc).
• At any iteration, a piece of the object is a scaled down, otherwise identical copy of the previous iteration (self-similar).

Mentor: These are the characteristics that Benoit Mandelbrot (who invented the term) ascribed to Regular Fractals in 1975.

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