When my evening was accidentally similar to Fawn Nguyen’s evening

Had a great time with our half group theory, half  Zometool project last night:


Just after I finished writing it up I saw these two posts on Twitter from Fawn Nguyen:


These two posts made me really happy.  First off, we both spent our night figuring out how to put a Zome cube inside a larger geometric structure.  Amazing!!

Even more exciting, though, the specific shape in Fawn’s pictures reminded me of a fun story from this summer.

Every year we head to Cape Cod with a bunch of friends from college – this year we had something like 10 adults and 10 kids in the house.  We knew ahead of time that one of the days was going to be rained out by the remnants of a hurricane and we needed to bring lots of extra indoor activities.  I brought our Zometool set as well as our 3D printer to have some math fun with the kids.

Some of the kids told me that they don’t like math, but since they’ve been on vacation with me lots of times they know that there will be math and fun together.  The 3D printer was a big hit, not surprisingly, and the Zometool set was even more of a hit.  We did a couple of activities out of this Zome Geometry book,

Book Pic

but mostly the kids just played.  One of the 11 year old girls who really does not seem to enjoy school math at all was particularly enthralled by the Zometool set.   She build some wonderful creations on her own including this one:


If you look at the 2nd picture posted by Fawn Nguyen above you’ll see a cube inside a shape known as a “rhombic dodecahedron.”  When you embed the cube inside of this shape you get to see 6 little yellow square pyramids on top of every cube face.  Definitely hard to see that a cube fits inside the rhombic dodecahedron at first, but as I talked about in yesterday’s blog post, the Zometool set is such a great aid because solving these problems is so much better when you can hold the shape in your hand.

Now look at the shape built this summer by the girl who doesn’t like her school math classes.  The important difference here is that the yellow pyramids are inside of the cube instead of on the outside.  Actually she’s divided up the top and bottom of the cube in a clever way so that the inside pyramid equivalent to the outside pyramid in Fawn’s example is built out of several smaller pyramids, but that is a minor detail for purposes of this blog post.  The really neat thing about her shape is that it shows you how to chop up a cube into 6 congruent pyramids, and that observation solves the problem posed in Fawn Nguyen’s first twitter post above!  The volume of the rhombic dodecahedron is exactly the volume of two cubes – the one on the inside plus the one formed by the 6 square pyramids on the outside.  That important second step comes courtesy of an 11 year old kid who doesn’t like math playing around with a Zometool set on a rainy summer morning.  Yay!!

So one shape comes from math teachers’ circle group in California, and the other from a kid just free building, for lack of a better phrase.  One shape from a group dedicated to teaching math, one from a kid who tells me that she doesn’t like math at all.  Funny how fuzzy the boundaries can be sometimes, and amazing how useful the Zometool sets are in helping people see math in a different, and probably more useful, way 🙂

A 3d Geometry project for kids and adults inspired by Kip Thorne

Yesterday I saw an incredible video on a discovery about black holes made by a team led by the physicist Kip Thorne.  The amazing thing about this discovery is that it came about through Thorne’s consulting work in the production of the new movie Interstellar.  You just never know when inspiration is going to strike!  The video I saw yesterday is here, though sorry I can’t get it to embed right:


What really struck me watching the clip above was the statement from Kip Thorne around 2:30 when he describes his reaction to seeing a picture of a properly rendered black hole:  “I’d known it intellectually, but knowing intellectually is completely different than seeing it, than feeling it.”

This thought was in my mind the rest of the day yesterday and I couldn’t stop trying to think up other math-related ideas that would be easier to understand if you were able to hold them in your hand.  At night I was flipping through my old Algebra book from college and found a great example by luck.  This book is one of my all time favorite math books – Artin was a terrific teacher:


In Chapter 7 of the book Artin explains some basic group theory and uses the rotation group of the icosahedron as an example.   One of the theorems shows that the group of rotations of the icosahedron is the same as second group – the alternating group A_5.  The details about A_5 aren’t important for this blog post, but what is important is that a critical geometric idea in the proof is that you can inscribe 5 cubes inside of a dodecahedron.  Here’s the picture Artin gives on page 200 of the book:


I never really completely understood this proof, and I never understood at all how these cubes inside of a dodecahedron related to the rotations of an icosahedron.  Thanks to Kip Thorne’s fun video and our Zometool set, though, I was determined to find out.  The kids were going to come along for the ride, too 🙂

To start off the project I asked the boys to boys to build a large icosahedron out of our Zometool set while I was at work today.  The sides of this icosahedron have length equal to two of the longest blue Zome struts.  Here’s our starting shape:

The dodecahedron is a dual shape of an icosahedron and it turns out there’s an amazing way to add a few new Zome struts to the icosahedron to make a shape that combines the icosahedron with a dodecahedron.  It is really cool to see this shape come together (another Zometool miracle!).  Here is the shape (and this + all of the rest of the videos are published in 1080p HD, so you can watch in hopefully non-blurry full screen to get a better view of the shapes):

With the icosahedron and dodecahedron together, now all we need to do to get to the shape Artin was describing in his book is to add in the cube.  I’m so happy that the Zometool set was able to help with this last step!  With this final shape built, we are holding the picture from the book (plus the original icosahedron) in our hand.  Again, hopefully the cube is visible in the video:

Now to see what the rotations of the icosahedron do to the cube.   Of course this is a fun fact all by itself, and that’s what the boys are seeing.  I’m seeing the critical step in showing that the Icosahedral group is isomorphic to A_5, though, which was the group theory piece that was so hard for me to visualize in college.   With the shape right in front of you it is easy to see how it all works – score one for Kip Thorne!

First we looked at what the rotations of the 20 triangles of the icosahedron do to the cube.  To make what we were rotating a little easier to see on camera we put some lego figures next to each vertex of the triangle we were rotating.  Hope that helps you to see the bottom triangle, and the rotations, a little easier in the video:

Next up, the 5 symmetries that come from the rotations of the icosahedron around a vertex.  As an aside, these 5 rotations sort of help you see the similarities between the icosahedron and the dodecahedron.  Again we lined up lego figures to help you see the rotations a little better in the video:

Finally rotating around and edge.  This is the easiest symmetry of the icosahedron to see since we are just rotating by 180 degrees around the middle of an edge.  It is interesting to see that although the icosahedron itself is unchanged by this rotation, the cube isn’t:

After we finished I asked the kids to help put away all of the Zometool pieces.  My younger son told me that he didn’t want to take apart “this awesome shape” just yet.  Yes!!

I’m glad the boys had as much fun with this project as I did.   I definitely got a better understanding of a piece of group theory that I never really properly understood before.  The boys were able to see (and build!) some amazing 3D shapes and also  play around with rotations and symmetries a little.  So so so much fun!!