Finding the Volume of an Icosahedron

Flipping through Zome Geometry last night I noticed that one of the exercises (Question 4 in section 17.2 to be specific) asked the reader to find the volume of an icosahedron.

This question struck me as being pretty difficult, but I was also intrigued. I started playing around and found that the Zometool set allowed you to answer the question in a surprisingly simple way. As an extra bonus, since we’ve been talking a little bit about pyramids lately this was a natural project to try out with the boys.

There are also a couple of other reasons why this project appealed to me. First, I’ve been trying to figure out how to incorporate a bit more computer math into our projects and Wolfram Alpha plays a big role in this one. My younger son definitely (and my older son probably) would have difficulty with the computations involved here. But since each kid is able to understand the geometric set up which really is the important piece of math in this project, I’m happy to leave the potentially confusing number crunching to Wolfram Alpha.

Second, I’ve just finished reading C├ędric Villani’s Birth of a Theorem. One of the takeaways for me from the book is that math projects usually involve pulling lots of different ideas together. This project pulls together ideas from pentagons, cubes, and pyramids to help find the volume of an icosahedron. Maybe not as many different ideas as you might need for Fields Medal winning research math (ha ha), but still there are some fun and surprising connections for kids to pull together here.

So, with that background we began the project by looking at pentagons. The goal here is to remind the boys of the relationship between the long blue and medium blue Zometool struts. By total coincidence we’d explored this problem recently here:

Walking the boys through a challenging problem form Solve My Maths

That lucky coincidence helped us get off to a good start:

In the prior project we actually worked through the calculations to find the Golden Ratio. Today we let Wolfram Alpha do that work for us:

Having reminded the boys of the relationship between the long and medium Zometool struts, we now built two different icosahedrons. One was hollow, and one had a star with red struts coming out from the center. The one with the red struts helps you to see that the icosahedron can be chopped up into 20 pyramids.

At the end of the last video my older son noticed that the height of the pyramid was going to be given by a yellow strut. This piece of information is one of the nice helping hands that the Zometool set gives us on this problem. We set out to see if we could find that height using the hint that the book gives – look at a triple size pyramid.

The other critical bit of help from the Zometool set is that the yellow struts form the long diagonal of a cube. That geometric fact tells us that the combined length of two long yellow struts is \sqrt{3} times the length of a long blue strut. This idea plus the golden ratio relationship between the longs and mediums allows us to calculate the height of our pyrmaid!!

Next we tried to go to Wolfram Alpha right away to calculate the volume of the icosahedron, but we got a little mixed up trying to remember all of the side lengths. So, we backed up and decided to write out the numbers on a whiteboard. This was actually my son’s suggestion at the end of the last video – should have listened! The work we do here helps us input the right numbers into Wolfram Alpha in the next video.

Now we are ready to go to the computer. Even with the numbers on the white board right in front of us the numbers are still sort of tricky to keep track up – you can imagine how messy it would be doing these calculations by hand. We eventually found an expression for the volume of an icosahedron!

The last step was looking up the actual formula on google to see if we got it right. Fortunately we did!

We end this video with a quick recap of the ideas that led us to the final solution.

So, a super fun project. It is pretty hard to imagine how you would approach this problem with kids without the Zometool set. With the Zometool set, though, the problem is not becomes approachable but it turns into a great example of mathematical reasoning. Wolfram Alpha comes in handy, too, since some calculations that would be pretty messy don’t bog down the project.

One more really fun morning thanks to Zome Geometry!