Exploring Calculus with 3d printing

Last week we were exploring volumes of various solids in the calculus course I’m teaching my son. That subject is a great opportunity to use 3d printing to enhance the course.

In fact, this old video from Brooklyn Tech about 3d printing was one of my first exposures to using 3d printing in math education:

Over the last week we printed 3 different shapes. Tonight I used these shapes as props to go back and review some of the volume concepts we’ve been studying. The ideas were new to my son and he hasn’t quite mastered all of them yet, so the review was productive. The videos are a bit longer than usual as we review some of the concepts, though.

Here’s the first shape: y = Sin(x) revolved around the x-axis:

The next set of shapes were not volumes of curves rotated around the x-axis, but rather shapes where the slices were squares:

Finally, we looked at the curve y = \sqrt{x - 1} from x = 1 to x = 5 rotated around the y-axis. Here my son remembered how to calculate this volume by looking at the slices parallel to the x-axis, but struggled a bit with when the slices were cylindrical shells – so we spent a long time on that 2nd part.

I’m happy that we have the opportunity to explore these shapes with our 3d printer – it definitely is incredible to be able to hold shapes like these in your hand!


Playing with Laura Taalman’s 3d printable “Scutoid”

Saw a great tweet from Laura Taalman over the weekend”

That shape was just “discovered” and is discussed on this New Scientist article:

oops – that tweet gives me a good picture, but the article itself is behind a paywall. ┬áHere are two free articles:

Gizmodo’s article on the Scutoid

The article in Nature introducing the shape

Last night I had the boys play with the shape (and I did not tell them what it was).

Here’s what my older son thought about it – sorry that it is a little hard to see the shape in the beginning. I add more light around 1:00 in:

Here’s what my younger son thought:

I thought it was interesting to hear that both boys thought that this shape would not appear in nature. I’ll have them

My talk at the 2018 Williams College math camp

[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:

Counting geometric properties in 4 and 6 dimensionsf

(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:

Kelsey Houston-Edwards’s hypercube video is incredible

One more look at the hypercube

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:

Here are the 2d slices of the 3d cube:

Here are the 3d slices of the 4d cube:

(4) Finally, we looked at the connection between cubes and combinatorics

I learned about this connection from this amazing video from Numberphile and Federico Ardila:

Here is the project I did with my older son after seeing Ardila’s video:

Federico Ardila’s Combinatorics and Higher Dimensions video is incredible!

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:

Jim Propp’s essay Time and Tesseracts

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.

3d printing totally changed my approach to talking about trig with my son

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

Screen Shot 2018-05-05 at 7.02.00 AM

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.

Playing with the nonagon tiles

Two of our recent project have involved studying a tiling of a nonagon from the book “Ernest Irving Freese’s Geometric Transformations”

Those two projects are linked here:

Using “Ernest Irving Freese’s Geometric Transformations” with kids

nonagon tiles

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.

Nonagon tiles

Last week we did a fun project using a pattern we say in “Ernest Irving Freese’s Geometric Transformations” by Greg N. Frederickson:

Using “Ernest Irving Freese’s Geometric Transformations” with kids

Screen Shot 2018-05-05 at 7.02.00 AM

I thought it would be fun to make some of the tiles – especially since my older son is studying trig right now. The tiles finished printing overnight:

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.

Sharing “developable surfaces” with kids thanks to a brilliant lecture from Heather Macbeth

[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:

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!