# Plane Figure Fractals

Shodor > Interactivate > Discussions > Plane Figure Fractals

 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. Do you remember what else we've discussed? 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's Triangle and Sierpinski's Carpet the area of the final figure was 0 - yet we could still see it. 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, 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. 