# 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!

# Having kids play with ” swarmalators”

Saw a couple of neat tweets on a new paper in Nature by Kevin P. Oâ€™Keeffe, Hyunsuk Hong, and Steven Strogatz:

It looked like playing with the “swarmalator” program would be a really fun way for kids to experience ideas from current math research even though the math underneath these results is a bit out of reach.

So, this morning we just played. Here’s how I introduced the ideas of the program – the two most important ones are (i) the strength of attraction of similar colors, and (ii) the strength of the desire for neighbors to have the same color (and both of these “strengths” can be negative):

After that short introduction I had my younger son (in 6th grade) play with the program to see what he found:

Next I let my older son (in 8th grade) play:

Finally, to talk about the ideas a bit more we went through 4 of the 5 examples at the bottom of the web page of the program we were using. I had the kids try to guess what was going to happen before we set the coordinates. Here are the first two examples:

Here are the last two examples – in this video the boys are getting the hang of how the program works and have several pretty neat things to say about what they are seeing (and what they expect to see):

We played with the program for about 20 min more after we turned off the camera. This program is definitely fun to play with and it was really fun to hear what the boys were guessing the various different states of the program would look like. Even with just two parameters, the kids really had to think hard to talk about the expected behavior. I think that lots of kids will really love playing around with this program.

# Thinking about a math appreciation class

Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:

I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:

Here’s a link to that set of projects:

Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)

So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic.Â  I chose folding and thought about what sort of ideas could be shared with students.

Below are 9 ideas that came to mind along with 30 second videos showing the idea.

(1) A surprise book making idea shown to me by the mother of a friend of my older son:

(2) Exploring plane geometry through folding

We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:

Our Patty Paper geometry projects

Here’s one introductory example showing how to find the incenter of a triangle:

(3) The Fold and Cut theorem

Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:

OUr Fold and Cut projects

Here’s one fun fold and cut example:

(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets

You can find Taalman’s post about these hinged polyhedra here:

Laura Taalman’s hinged polyhedra blog post on her Makerhome blog

And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!

Which comes from this amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

[space filled in with random words to get the formatting in the blog post right ðŸ™‚ ]

(5) An amazing cube dissection made by Paula Beardell Krieg

We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:

Our projects based on Paula Beardell Krieg’s work

(6) And Paula didn’t just stop with one cube ðŸ™‚

(7) Two more of Laura Taalman’s prints

Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.

Here’s Taalman’s blog post about the Peano curve:

Laura Taalman’s peano curve 3d print

(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke

As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:

(9) Current research by Laura DeMarco and Kathryn Lindsey

Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.

Quanta Magazine’s article on DeMarco and Lindsey’s work

The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids

So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.

# 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.

# Using 3d printing in the college classroom

I saw a couple of tweets from Steven Strogatz yesterday that got me thinking about how you might use 3d printing in the college classroom:

The last tweet, in particular, made me think that having the 3d print versions of the two shapes would be useful. Before I get too far in to this post, though, I had to throw this post together pretty quickly to be able to fit in an hour of shovelling prior to heading to work! Sorry if it isn’t the most well-written or well-argued post. The main takeaway I want is that I think there are many great uses for 3d printing in the college math classroom.

The topic of 3d printing and calculus is one that I’ve thought about briefly before – see these old posts:

3d Printing and Calculus Concepts for kids

Using 3d printing to explore some basic ideas from calculus

Here are the shapes from the first post linked above – I think they would help students understand ideas like Riemann sums and volume by slicing:

Here are the two 3d shapes from the second Strogatz tweet from yesterday. Unfortunately we lost power in the middle of the night before the print project was complete, but you’ll get the idea. One of the things that comes through immediately in the prints is the difference in size of the two shapes:

Finally, an important shape from advanced algebra – a cube inside of a dodecahedron. This shape appears (and plays an important role) in Mike Artin’s Algebra book:

I found it hard as a student to understand the shape solely from the picture. Holding the shape in my hand, though, makes it much easier to see what is going on (I have made the cube slightly larger to highlight it):

So, while I’m sure it is true that learning to draw some of these shapes by hand is useful, I also think that 3d printing can be an important tool to help students see, understand, and experience the same shapes.

# Steven Strogatz’s circle area project – part 2

Yesterday we did a really fun project inspired by a tweet from Steven Strogatz:

Here’s tweet:

Here’s the project:

Steven Strogatz’s circle-area exercise

During the 3rd part of our project yesterday the boys wondered how the triangle from Strogatz’s tweet would change if you had more pieces. They had a few ideas, but couldn’t really land on a final answer.

While we punted on the question yesterday, as I sort of daydreamed about it today I realized that it made a great project all by itself. Unlike the case of the pieces converging to the same rectangle, the triangle shape appears to converge to a “line” with an area of $\pi r^2$, and a lot of the math that describes what’s going on is really neat. Also, since my kids always want to make Fawn Nguyen happy – some visual patterns make a surprise appearance ðŸ™‚

So, we started with a quick review of yesterday’s project:

The first thing we did was explore how we could arrange the pieces if we cut the circle into 4 pieces.

After that we looked for patterns. We found a few and my younger son found one (around 4:09) that I totally was not expecting – his pattern completely changed the direction of today’s project:

In this section of the project we explored the pattern that my son found as we move from step to step in our triangles. After understanding that pattern a bit more we found an answer to the question from yesterday about how the shape of the triangle changes as we add more pieces.

Both kids thought it was strange that the shape became very much like a line with a finite area.

The last thing that we did was investigate why the odd integers from 1 to N add up to be $late N^2$. My older son found an algebraic solution (which, just for time purposes I worked through for him) and then we talked about the usual geometric interpretation.

So, a great two day project with lots of fun twists and turns. So glad I saw Strogatz’s tweet on Friday!

# Steven Strogatz’s circle area exercise

Saw this really neat tweet from Steven Strogatz yesterday:

I asked him if he had downloadable versions of the circle and he was nice enough to share the templates with me (yay!)

So, with a little enlarging and a little cutting we had the props ready to go through the exercise.

We began with a short conversation about circles. My older son knows lots of formulas about circles from his school’s math team practices, but my younger son doesn’t really know all of the formulas. The quick review here seemed like a good way to motivate Strogatz’s project:

Now we moved into Strogatz’s project – how do we show that the area of a circle is $\pi * r^2$? We cut the circle into the 16 sectors and rearranged them into a shape that was more familiar to us:

Next was the big challenge and the really neat idea in Strogatz’s first tweet – there is a different shape we can use to find the area. The boys were able to find this triangle fairly quickly, but then we had a really fun discussion about what the triangle would look like if we used more (smaller) sectors. So, the surprising triangle from Strogatz’s tweet led to a really fun and totally unexpected discussion! It is so fun to hear kids think through / wonder about math questions like the one they asked about the new triangles.

The last part of the project today was inspired by a tweet from our friend Alexander Bogomolny that was part of the thread Strogatz’s tweet started on Twitter yesterday:

I love it when Twitter writes our math projects for us ðŸ™‚

I had the kids look at the picture and describe what they saw. At the end I asked them why they thought the slanted lines in the triangle were lines and not curves – they had interesting thoughts about this little puzzle:

The amount of great math shared out twitter never ceases to amaze me. Thanks (as always!) to Steven Strogatz and to Alexander Bogomolny for inspiring this project about circles. Can’t wait to try out this project with other kids.

# Sharing math with the public and especially with kids

My wife and kids are up hiking in New Hampshire this weekend and I’m home with a cat who misses the kids. Yesterday I was watching Ed Frenkel’s old Numberphile interview about why people hate math:

The line about 50 seconds in to the video has always really resonated with me – “How do we make people realize that mathematics is this incredible archipelago of knowledge?” As has the his point later in the video that mathematicians have not generally done a great job sharing math with the public (say from 5:00 to 6:30).

Frenkel’s piece has played a role in many of my blog posts, here are three:

Sam Shah – a high school teacher in New York – wrote a great piece about sharing math that is not typically part of a high school curriculum with kids, and gave some suggestions for projects:

A Partial Response to Sam Shah

Lior Pachter wrote an incredible blog post about sharing unsolved math problems under the Common Core framework. I copied his idea but used math from mathematicans rather than unsolved problems:

Sharing math from Mathematicians with the Common Core

Then when the sphere packing problem was cracked by Maryna Viazovska earlier this year, I wrote about how this was a great opportunity for mathematicians to share a math problem with the public:

A challenge for professional mathematicians

As you can tell, I watch Frenkel’s video quite a bit ðŸ™‚ While I was watching the video yesterday I received this message:

Though he isn’t a professional mathematician, this article from Brian Hayes is really close to what I’d love to see from mathematicians.

As are the articles by writers like Erica Klarreich and Natalie Wolchover at Quanta Magazine:

Quanta Magazine’s math articles

and mathematician Evelyn Lamb who has somehow found the time to write more than 150 articles on her “Roots of Unity” blog for Scientific American:

Evelyn Lamb’s blog on Scientific American’s website

There were probably at least 10 to pick from, but here’s an example of how I’ve used one of Lamb’s pieces to talk a little bit about topology with my kids:

Using Evelyn Lamb’s “Infinite earring” article with kids

So, with that all as introduction (!) I was very excited to see Steven Strogatz share an article from Rich Schwartz last night:

I really enjoyed our project with Schwartz’s “Really Big Numbers” and I’m happy to see that he’s writing more about sharing math with kids. Hopefully Schwartz’s article will inspire a few more mathematicians to share some fun math with kids (or with the public in general). I’d love to expand this list of projects beyond 10 ðŸ™‚

# A fun coincidence with our “1/3 in binary” project

Saw this tweet from Matt Henderson (via a Steven Strogatz retweet) today:

It first reminded me of one of Patrick Honner’s blog posts from a few years ago:

Honner’s post plus a lucky coincidence with a Numberphile video inspired a fun project with the boys:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s square

Revisiting 1/3 in Binary

Since I was looking for a quick little something to do with the boys tonight, I head each one of them take a look at the tweet and tell me what they thought.

My younger son went first – he had lots of neat thoughts about the shape and we eventually found our way to the connection between this shape and writing 1/3 in binary:

My older son went next – he didn’t see the connection right away, but we eventually got there, too (oh, and ugh, just listening to this video I realize that I misunderstood my son when he was talking about a geometric series – whoops, he did say the right definition).

Thank you internet – what a fun coincidence!

# Using the Mathematical Etudes videos with kids

Saw this tweet from Steven Strogatz earlier in this month:

The videos are amazing and I finally got around to using one of them with the boys last night. We looked at one of the paper folding videos here (I’m sorry that I don’t know how to embed this video):

Mathematical Etudes – Origami “Single Cut”

I picked this project because we had previously done several “one cut” projects after seeing the amazing video from Katie Steckles and Numberphile:

Here are the three projects that we did after seeing Steckles’s video:

Our One Cut Project

Fold and cut project #2

Fold and cut part 3

After watching the “Single Cut” Mathematical Etudes video with each of the boys separately, we tried to cut out a triangle.

I did the project with my older son first. He remembered how to make angle bisectors by folding the paper, so that part went quickly. He skipped drawing in the perpendiculars (!) and went straight to the folding part and was able to get the folds just about right. Pretty efficient – ha!

My younger son had a little more difficulty recreating the procedure in the video, but we did get there. He used a protractor to find the angle bisectors and then drew in the perpendiculars by sight. The folding was more difficult for him which surprised me a little, but I guess the dexterity required for this folding is easier for a 6th grader than for a 4th grader.

I’m sorry this video is nearly 10 minutes long, but I hope it shows that even a fairly young kid can use this Mathematical Etudes video to learn to cut out a triangle in one cut!