# Playing with 3d printed versions of shapes theorized by Hermann Schwarz

Saw a neat tweet earlier today about 3d printing, math, and engineering:

I recognized some of the shapes in the article as ones that we’d played with before:

The grey shape displayed in the article is a “made thicker for 3d printing” version of the surface $\cos(x) + \cos(y) + \cos(z) = 0.$ I thought it would be fun to print that shape today and use it for a little project with the kids tonight. Here’s the Mathematica code and what the print looks like in the Preform software:

8 hours later the print finished and I asked the boys to describe that shape plus the gyroid. It is always fascinating to hear what kids see in unusual shapes. My younger son went first:

Here’s what my older son had to say (and he’s starting to study trig, so we could go a tiny bit deeper into the math behind the shape I printed today):

Next we watched the video about the shapes made by Rice University:

After watching the video I asked the boys to talk about some of the things they learned:

Of course, mostly they didn’t want to talk about the shapes – they wanted to stand on them! So much for an 8 hour print and 45 min of trying to clean out the supports . . .

Here’s how the standing went:

Definitely a fun project and a fun way to show kids a current application of both theoretical math and 3d printing!

# Sharing Kelsey Houston-Edwards’s topology video with kids

Kelsey Houston-Edwards’s latest video is terrific:

This one is particularly easy to share with kids because there are several puzzles where she asks you to stop and think about the solution. I began the picture frame puzzle as the starting point for our project today.

The puzzle goes roughly like this:

A common way to hang a picture is to use two nails in a wall and run the wire around those two nails. Assuming the nails / wall are strong enough, if you remove one of the nails the picture will still hang. Is there a way to hang a picture with two nails so that if you remove either of the nails the picture will fall?

We took a shot at this puzzle using yarn and snap cubes. It was a good challenge for the boys:

In the last video we got the picture to fall once, but the boys weren’t quite clear what happened – but now they at least knew it was possible! Here we explored the idea more carefully:

Next we finished watching the video and then discussed what we saw (as I publish this post the video preview isn’t embedding properly, but is really just audio anyway):

Finally we looked at two sets of shapes that appeared in the video that we’ve looked at before. The first is a 3d print of Henry Segerman’s “Topology Joke” and the 2nd is a set of “rollers” that we’d made after seeing a Steven Strogatz tweet. The tweet and the roller project are here:

3d printing and rollers

Another fun project from Kelsey Houston-Edwards’s amazing math series. Sorry to be brief on this project, but I had to get this one out quick because of a bunch of activities going on today.

# Things to print and do in the 4th dimension

Today our math and 3d printing project combined ideas from two great books.  First Matt Parker’s book Things to Make and Do in the Fourth Dimension and Henry Segerman’s book Visualizing Mathematics with 3D Printing

We started out the project today by watching Parker’s fun video about 4 dimensional platonic solids:

Next we look at some of the 3d prints we have of projections of the four dimensional platonic solids from Segerman’s book. Here’s what the boys had to say:

Then we went through some of the shapes in more detail. Here’s the 5-cell

Here’s what the boys thought about the two different versions of the hypercube that we have.

I’d add that our Zome version of Bathsheba Grossman’s “Hypercube B” blew me away, too:

Finally, we talked about the 24-cell and the 120-cell. Sorry this part went a little long, but the shapes are really cool!

I’m loving 3d printing more and more every day. The opportunities to take ideas from books and videos and put them directly into the hands of kids is just amazing. Thanks to Parker and Segerman for doing the heavy lifting for me on this project!

# Playing with some 3d printed knots

Today we looked at some 3d printed knots designed by Laura Taalman and Henry Segerman.

Two are versions of Taalman’s “rocking knot” which we found here:

Laura Taalman’s Makerhome blog: Day 110 – the Rocking Knot

The second is the Torus knot from Segerman’s new book Visualizing Mathematics with 3D Printing.

We started the project today by just talking about the knots. Comparing the two knots that are actually identical was useful in refining the language they used to talk about knots.

Next they wanted to try to compare the two identical knots by looking at their crossings. My older son had the idea of assigning a +1 to every “over” crossing and a -1 to every “under” crossing. My younger son noticed that this counting method should always produce a net 0 because we counted the over and under crossing for each crossing exactly once.

New we tried to compare Segerman’s torus knot to Taalman’s rolling knot. Here we used the “tangle” from Colin Adams’s book Why Knot?

One fun thing that came up by accident in this video is an amazing shadow cast by Taalman’s knot – that was a really fun surprise.

Unfortunately, it proved to be a bit difficult to get the tangle back together so we had to pause the video at the re-connect the tangle off camera. It is really neat, though, to watch kids try to make a copy of a knot.

Once we got the tangle connected we started the next video. Since the tangle can move around, it isn’t that hard to manipulate the tangle from the form Segerman’s knot to the form of Taalman’s knots. In fact, it happened more or less by accident!

As I mentioned above, it is actually a pretty difficult task for the kids to describe the features of the knots when they compare them – even with a knot as simple as the trefoil knot. I think one of the neat parts of this particular project is working on using more precise mathematical language.

So, a fun project. We have a new 3d printer and I’m really excited about using many more 3d printing ideas from Taalman and Segerman to explore math with the boys.