[had to write this in a hurry before the family headed off for a vacation – sorry that this post is likely a little sloppy]

Yesterday I gave a talk at a math camp for high school students at Williams College. The camp is run by Williams College math professor Allison Pacelli and has about 20 student.

The topic for my talk was the hypercube. In the 90 min talk, I hoped to share some amazing ideas I learned from Kelsey Houston-Edwards and Federico Ardila and then just see where things went.

A short list of background material for the talk (in roughly the order in the talk is):

(1) A discussion of how to count vertices, edges, faces, and etc in cubes of various dimensions

This is a project I did with my kids a few years ago, and I think helps break the ice a little bit for students who are (rightfully!) confused about what the 4th dimension might even mean:

(2) With that introduction I had the students build regular cubes out of the Zometool set I brought. Then I gave them some yellow struts and asked them to construct what they thought a hypercube might look like. From the prior discussion they knew how many points and lines to expect.

To my super happy surprise, the students built two different representations. I had my boys talk about the two different representations this morning. Funny enough, they had difference preferences for which was the “best” representation:

Here’s what my older son had to say:

Here’s what my younger son had to say:

At the end of this section of my talk I showed the students “Hypercube B” from Bathsheba Grossman (as well as my Zometool version):

(3) Now we moved on to looking at cubes in a different way -> standing on a corner rather than laying flat

I learned about this amazing way to view a cube from this amazing video from Kelsey Houston-Edwards. One of the many bits of incredible math in this video is the connection between Pascal’s triangle and cubes.

Here are the two projects I did with my kids a after seeing Houston-Edwards’s video:

After challenging the kids to think about what the “slices” of the 3- and 4-dimensional cubes standing on their corners would be, I showed them the 3D printed versions I prepared for the talk:

Part of my talk on hypercubes at Williams tomorrow -> 3D slices of a hypercube standing on a corner produce tetrahedrons and an octahedron pic.twitter.com/f7s8XM6kSB

I walked the students through how the vertices of a square correspond to the subsets of a 2-element set and then asked them to show how the vertices of a cube correspond to the subsets of a 3-element set.

There were a lot of oohs and ahhs as the students saw the elements of Pascal’s triangle emerge again.

Then I asked the students to find the correspondence between the 4-d cubes they’d made and subsets of a 4-elements set. I was incredibly happy to hear three different explanations from the students about how this correspondence worked – I actually wish these explanations were on video because I think Ardila would have absolutely loved to hear them.

(5) One last note

If you find all these properties of 4-D cubes as neat as I do, Jim Propp has a fantastic essay about 4 dimensional cubes:

By lucky coincidence, this essay was published as I was trying to think about how to structure my talk and was the final little push I needed to put all the ideas together.

For the last two weeks we’ve been playing with this book:

Our most recent project involved one of the pentagon dissections. My son wrote the code to make the shapes on his own. We use the RegionPlot3D[] function in Mathematica. To make the various pieces, he has to write down equations of the lines that define the boundary of the shape. Writing down those equations is a fantastic exercise in algebra, geometry, and trig for kids.

Here’s his description of the shapes and how he made the pentagons:

Next we moved on to talking about one of the complicated shapes where the method he used to define the pentagon doesn’t work so well. I wish I would have filmed his thought process when he was playing with the code for this shape. He was really surprised when things didn’t work the first time, but he did a great job thinking through what he needed to do to make the shape correctly.

Here is his description of the process followed by his attempt to make the original shape (which he’d not seen in two days . . . )

I’m so happy that he’s been interested in making these tiles. I’ve honestly never seen him so engaged in a math project. The original intention of this project was just for trig review, but now I think creating these shapes is a great way to use 3d printing to introduce basic ideas from trig to students.

After school yesterday I had each of the boys make a pattern with the nonagon tiles and then build the two patterns that were in the book. The videos below show there work. My younger son went first:

Here’s what my older son had to say:

This project was super fun from start to finish. Hearing the thoughts from the boys after seeing the pattern initially was really fun. Building and printing the blocks was a nice geometry / trig lesson. Then having the boys play around with them made for a really satisfying end to the project. I hope to do more like this in the near future.

Last night my son and I talked about how you could make these tiles, with a focus on the trig and algebra required to define the shapes.

Here’s the introduction to the topic:

Now we talked about how to define the kite shape in the tiling. This involves talking about 40 and 50 degree angles:

Finally, we talked through the last part – finding the final point is pretty challenging. Turns out, though, that we don’t have to find the coordinates of the point because we can write down the equation of the top line pretty easily:

I’ve been happily surprised that 3d printing is a fun way to help kids explore 2d geometry. I’m excited to have my son try to make some other tiles from the book on his own for our next project.

[This is a redo of a blog post from January 2018 that somehow ended up 1/2 deleted. Not sure what I did to that old post, but I didn’t want to lose the ideas.]

In January 2018 I attended a terrific public lecture given by Heather Macbeth at MIT. The general topic was differential geometry, and the specific topic she discussed was “developable surfaces.”

Here’s an example from the talk:

Earlier today I attended an absolutely fantastic public lecture given by Heather Macbeth of MIT. The topic was "developable surfaces" -> surfaces that can be made from a sheet of paper. Can't wait to share this topic with kids: https://t.co/SjSbGfWxrq#math#mathchat

I also printed a few examples and shared them with the boys the next day:

Here’s what my older son had to say about the shapes:

Here’s what my younger son thought:

These are really neat surfaces to explore. If you look at some of the mathematical ideas for “Developable surfaces” you’ll find that some of the surfaces are actually pretty easy to code, print, and share with kids!

[sorry at the beginning that this post feels a little rushed. I wrote it during an archery class my son takes, but I forgot the power cord to my laptop and only had 20% battery at the start . . . . ]

Over the last week I saw two really neat videos from Annie Perkins on the Cairo pentagon tiling:

Today I did a project with my younger son with 3d printed versions of the pentagons that I made today (after a few glorious fails . . . .). Sorry that the tiles don’t show up super well on camera when they are pushed together – I’d hoped that the white background with show through the gaps, but not so much π¦

Before starting I showed my son the two videos from Perkins and began the project by asking him to try to recreate the shapes he saw. He liked the tiling but ran into a little trouble trying to recreate it. It turns out that tiles also fit together in a way that doesn’t extend to a tiling of the plane. My son had a nice geometric explanation about why the shape he found wouldn’t extend to the full plane.

After running into a little difficulty in the last video, he started over with a new strategy. That new strategy involved putting the tiles together in groups of two and fitting those groups together. This method did lead to a tiling that he thought would extend to the full plane.

Definitely a fun project. You can see some links to other tiling projects we’ve done in yesterday’s project with my older son. Tiling is definitely a topic you can have a lot of fun with on a few different levels – from younger kids talking about the shapes they see, to older kids learning how to describe the equations of the boundary lines and coordinates of the points. Making the tiles is a fun 3d printing project, too.

I thought it would be a fun idea to add to the list of our growing list of pentagon projects. At this point I’ve lost track of all of them, but they got started with this amazing tweet from Laura Taalman:

After seeing Perkins’s tweet I started down the path of making the Cairo tiling pentagons but super unluckily had a typo in my printing code. At least my cat made good use of the not-quite-Cairo pentagons:

So, while I wait for the correct pentagons to print, I thought I’d talk about the special shape of the Cairo tiles with my older son. One of the neat things about all of these pentagon projects is getting to talk about geometry with kids in sort of non-standard, non-textbook way. Tonight’s conversation was about coordinate geometry using the properties of the Cairo pentagon.

Here’s a pic from the Wikipedia page on the Cairo tiles:

To start the project I drew the shape on our board and asked my son to find the coordinates of the points. This is a bit of an open ended question because you have to know the lengths of the side so know the coordinates – I was happy that he noticed that problem (and, just to be 100% clear, I don’t know for sure if there are restrictions on the sides for the Cairo tiling – I’ll learn that when the new pentagons finish printing – ha ha).

Here’s how he started in on the problem:

For the second part of the project he had to make one more choice for a side length, and then he was able to find the coordinates of all of the corners of the pentagon.

One of the great (and happy) surprises with math and 3d printing is that you get neat opportunities to explore 2d geometry. Some of our old projects exploring 2d geometry with 3d printing are here:

Over the winter break I began to think about collecting some of our 3d printing projects into to one post to highlight various different ways that 3d printing can be used to help kids explore math.

The post got a little long, but if you are interesting in thinking about 3d printing and math, hopefully there are ideas in here that either catch your eye.

(1) Archimedes’s proof relating the volume of a sphere, a cone, and a cylinder

I asked my younger son to pick his favorite 3d printing exercise – here’s what he picked:

My older son’s favorite project involved the rhmobic dodecahedron:

We’ve actually done a bunch of projects – both 3d printing and Zometool projects – with the rhombic dodecahedron. Here’s a link to all (or probably most) of them:

This is probably my favorite 3d printing project that we’ve done on our own. I didn’t do a specific project with the boys using the shape because it is really fragile (in fact, I have 3 other broken ones . . . ).

The problem is -> can you cut a hole in a cube large enough so that you can pass another cube of the same size through the first cube?

An old project where we talk about the problem (without 3d printing) is here:

The boys had really enjoyed trying to solve Iwahiro’s puzzle (which may be more difficult to get apart than it is to put together!).

(6) The Gyroid and other minimal surfaces

3d printing allows you to explore some incredible shapes. For instance:

Rice University scientists have successfully 3D printed Hermann Schwarz' Schwarzites: theorized minimum surfaces having negative Gaussian curvature. https://t.co/hEpVK0GH7M@RiceUniversity@RiceUNews

Another calculus-related project is here, and it includes a great video from Brooklyn Tech that helped show me the possibilities 3d printing had for helping kids explore math:

Here’s a really fun shape to play with – the rattleback. It wants to rotate one way, but not the other way. There’s very little indication when you look at it that it would have such an odd property:

(10) James Tanton’s tetrahedron problem

This one has a special place in my heart because it was one of the first times we used 3d printing to solve a “new to us” problem. I loved how these shapes came together. The problem involved understanding the locus of points that were 1 unit away from a tetrahedron:

We’ve done a bunch of projects related to the 4th dimension that have been aided by 3d printing. Most of this work has been inspired in one way or another by Henry Segerman. Here are a few examples:

Another one of my all time favorites projects came from Laura Taalman. Right after the discovery of a 15th type of pentagon that tiles the plane, Taalman created 3d print models of all 15 of the pentagons so that anyone could explore this new discovery:

This is one of the most amazing illusions that you’ll ever see π

Thanks to @landisb for suggesting I 3D print my impossible cylinder. And thank you to her for printing my file. I will have to try others. pic.twitter.com/X9Drl3Il4w

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

Rice University scientists have successfully 3D printed Hermann Schwarz' Schwarzites: theorized minimum surfaces having negative Gaussian curvature. https://t.co/hEpVK0GH7M@RiceUniversity@RiceUNews

The grey shape displayed in the article is a “made thicker for 3d printing” version of the surface 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:

Generating an approximation to the shape isn't that difficult (and might be a fun intro 3d printing exercise for a Trig class). Cleaning up the final prints and getting rid of the supports is going to be a bit of a headache, though. pic.twitter.com/YSTRs3wx5Y

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!