## James Tanton’s geometry problem and 3D Printing

This past winter we saw this really amazing video (posted by Patrick Honner)  about how students were using the 3D printer at Brooklyn Tech:

Inspired by this video we decided to get our own 3D printer and see what we might be able to do with it.  The first lesson was easy – follow Laura Taalman’s 3D printing  blog:

http://makerhome.blogspot.com/2014_01_11_archive.html

From following her blog we saw incredible example after incredible example of how 3D printing can help open math doors that weren’t so easy to open before.  Using what we learned from Taalman’s blog, the first project we tried on our own was about the Prince Rupert Cube.  We wrote about some of the fun we were having here:

https://mikesmathpage.wordpress.com/2014/03/12/learning-from-3d-printing/

The second project that we did on our own came from a project that James Tanton posted on Twitter last week:

The problem itself is a little too advanced for the kids, we haven’t really discussed geometry at all,  but just trying to understand the shape of this object proved to be a fascinating exercise.  Again, using what I’ve learned from Taalman’s blog, I made a sequence of 5 shapes on Mathematica that help you build the shape up in your mind:

After a little bit of travel for everyone in the family this week we were finally able to sit down this morning and talk through Tanton’s problem.

In the first video I introduce the kids to the problem and talk through some of basic geometric ideas in it.  What is a tetrahedron?  What is surface area?  What does it mean that the area of each face is 1?  The nice thing about this problem is that the basic ideas are accessible to kids:

In the second video we take a quick look at a slightly easier problem – what if we were trying to find the points in a plane with a distance 1 from an equilateral triangle?  This just came up on the fly, but it turned out to be a really valuable detour since it gave us some nice insight into the 3D shape.    In fact, after reviewing this video I may come back to the 3D problem tomorrow to show how the pieces of the spheres come together since it is so similar to what we discussed in the 2D case.

Next we moved on to the 3D printed shapes that I made this week.  I think being able to hold these shapes in your hand really helps to get your head around Tanton’s problem.  As explained in the Brooklyn Tech video, learning how to write code that produces these shapes is a great math activity all by itself.  This would be a super fun project for students learning about linear algebra, 3D geometry, or even just learning to program in Mathematica.    I also think that walking through these 5 shapes we printed is a nice example of breaking complex problems down into simpler ones to help you get to the solution:

All in all, a really fun exercise and a neat example of how 3D printing can help understand some fun geometry problems.  Thanks to James Tanton for posting this great problem.

1. I love this – and what a great use of 3D printing – very exciting!
Just two things:
1. Did you really not print the final model?? That really is process over product!
2. To me it looks like it would have been better if James Tanton had asked the same question of a cube with unit face area. The fourth root stuff had to be skipped over anyway. The cube is simpler, but seems to me to involve all the same learning for kids.

2. Hi Simon –

A big contributing reason for not printing the final product was my own carelessness. I didn’t save the Mathemtaica worksheet and then my kids closed all the open applications in order to play Minecraft. Whoops. I was able to recreate everything up to what I printed last night, but the last set of rotations to produce the remaining three cylinders wasn’t working right. Couldn’t find the typo and it was already midnight so I just printed what I had.

On the second point, if you go back a week on Tanton’s twitter feed, he actually asked a series of questions. Don’t remember if he asked about the cube or not, but it was an interesting set of questions. I’d love to know where the requirement that the triangles have area 1 came from. It does make for one extra little bit of interesting work as you code, though, so for a student thinking about the problem I think it is actually a nice thing to think through.