It was my younger son’s turn to pick a topic for Family Math and last night he told me that he wanted to talk about fractals. We’ve talked a little bit about fractals over the years and also a little bit recently, too. Dan Anderson’s neat Mandlebrot Set program, for example, is something that we spent time playing around with in September:
Since we were a little short on time this morning I wanted to talk about a simple (and fun) fractal and chose to use the Koch Snowflake as the main example. Some of our previous talks about the Koch Snowflake have felt a little rushed since I was trying to show two neat properties at the same time: (i) the perimeter grows to infinite, and (ii) the area stays finite. This time around I decided to do a slightly deeper dive into just the first property and illustrated the second property by simply drawing a larger circle around our shape at the end of the 2nd video. Focusing only on the perimeter also gave me the idea to use one of Laura Taalman’s 3D printing projects to illustrate how an infinitely long set of line segments can even fill up a finite area. That part (the 3rd video) only really scratches the surface of the topic, but holding the objects in your hand and seeing how they stretch out does leave quite an impression. Taalman’s 3D printing project for the Peano Curve is here:
We began our project with the boys talking about what they thought fractals were. After that introduction we drew the first few iterations of the Koch Snowflake. The goal of this part of the project was to count the sides and calculate the perimeter at each step. It was fun to hear the different ways that the boys thought to calculate the perimeter. After the first iteration, for example, my older son saw that there were 12 total sides by noticing that there were 3 groups of 4, while my younger son noticed that there were 6 groups of 2 sides by looking at the corners of the snowflake.
The next part of the talk goes into a little more detail about the number of sides and the perimeter. For the first couple of minutes we talk about the number of sides in a little more depth. The boys seemed to understand why the number of sides was being multiplied by 4 every time, and even seemed to be able to explain that idea using geometry. It was nice to see them get a little bit better understanding of this idea.
Next we talked about the perimeter and how it changes at every step. Unfortunately I had a couple of stumbles talking through this part, but hopefully those little stumbles don’t totally confuse the message. We calculate the perimeter for the first few stages and show that it keeps growing by a larger and larger amount every time. That idea leads us to believe that the perimeter will eventually become infinite. However, since the shape itself seems to always be contained inside of a larger circle, the area seems to remain finite. The boys use the terms “infinitely finite” and “finitely infinite” to describe this seemingly odd situation 🙂
The concept of a shape with infinite perimeter but finite area seems pretty strange when you hear about it for the first time (or maybe even the first several times!). To help get a better feel for this situation I used the 3D printing project from Laura Taalman I mentioned above. She showed how to print an iteration of the Peano Curve – the first example of a space filling curve discovered by the Italian mathematician Giuseppe Peano in 1890. You can read a little bit more about this curve here:
For the purpose of our little talk this morning, the important feature of this curve is that it gets infinitely long while filing up a finite area. The neat thing about having the 3D printed version is that we can actually pull the curve apart and make a straight line again (well, nearly a straight line). Although this is not exactly the same idea as an infinite perimeter enclosing a finite area, I think that holding the Peano curve in your hand and watching it unfold into a straight line helps you get our head around how an infinite perimeter could enclose a finite area.
So, a quick morning, but we still got to see some fun concepts from fractal geometry. Going into a bit more depth than usual on the idea of infinite perimeters and finite areas was fun. Also, once again the 3d printer helps us gain a better understanding of some geometry, though this time it is 1 and 2 dimensional geometry, which is kind of a nice surprise.