A Zometool Koch Snowflake

We have a bit of an unusual opportunity right now for some fun Zometool projects since we have a living room with no furniture. Last week we made a fun Zometool “snowman” based on a chapter in our Zome Geometry book.

Snowman here:

Project write up here:

A little Zome geometry in our new house

Today we kept with the winter theme and made a giant Koch Snowflake:

We started with an equilateral triangle having 27 large blue struts on a side. We originally wanted to make a triangle with 81 struts per side, but the room wasn’t large enough!

Once we finished the construction I had the boys talk about how they built it and also about the area and perimeter of the shape:

The next step is to divide the sides of the equilateral triangle’s sides into thirds and then build new equilateral triangles on top of the middle thirds. It was interesting to hear them talk about the perimeter and area of this new shape. I intentionally didn’t have them write anything down – I just wanted to hear what they had to say and specifically hear how their geometric intuition helped them understand these shapes.

In the next step the area and perimeter are a bit more difficult to calculate, though the shape is starting to look pretty cool. They have a bit of trouble talking through the area and perimeter calculations, but we eventually get through these calculations. One of the things giving them trouble is that you add two new sides to the shape for each new triangle, but you also have to take away one side. I think physically building the shape helped them understand this addition and subtraction a bit better.

The last step gets really tough and I had to help them see the pattern. They are looking for an arithmetic pattern, but the pattern here is geometric. Once I ask them about multiplication they seemed to be able to understand the pattern.

I split the perimeter and area calculations into separate movies. The main idea here is that the perimeter is getting larger by a larger amount each step. That fact tells us that the perimeter is going to infinity!

The area, though, doesn’t seem to be going to infinity. After all, it can still fit inside of the living room! My older son uses the term “infinitely finite” for this situation π

Here’s the perimeter discussion:

and here’s the area discussion:

So, a fun project bringing together lots of different mathematical ideas. There’s some nice arithmetic practice here, some great work spacial relationships ( building this shape isn’t super hard, but it isn’t as easy for kids as I thought it would be either), and of course the fun work with fractals and infinity. Nice way to start the morning π