Student: Just what is self-similarity?? Is it when you repeat the same steps over and over, like in iteration and recursion?

Mentor: It is related to that. However, iteration and recursion describe the process of repeating steps. Self-similarity is a property of the object, not of the steps used to build the object. Think of it this way: When we look at stage 2 of the Hilbert curve, we can find a copy of stage 1:

This copy is a smaller scale version of the original. We can find this in the Koch Snowflake, too:

Can you find self-similarity in the other fractals?

Student: I see it in zooming on the Tortoise and the Hare and Another Hilbert Curve. I'm still having trouble understanding why self-similarity is different from iteration and recursion, though.

Mentor: It is difficult to explain. Now that you understand what self-similarity is, it may help to think of it this way: iteration and recursion frequently result in self-similar objects.

Student: So, the Hilbert Curve is a self-similar object, but the way we construct it is an iterative process. I think I understand, now.