# Revisiting the angle sum arctan(1/2) + arctan(1/3)

Today we did a 3d printing project revisiting an angle sum that we’d looked at last week -> arctan(1/2) + arctan(1/3).

We started by reviewing how to approach the sum using complex numbers:

Next my older son explained a geometric way to approach the problem:

Now we went to Mathematica to create the 4 triangles using the RegionPlot3D function. It is a nice geometry exercise to have kids describe the boundary of a simple 2d object:

At the end of the day I had my younger son use the shapes to assemble the 3×2 rectangle and describe how this arrangement showed that the original angles added up to 45 degrees:

I like using 3d printing to help kids see math in a different way. The problem today was originally inspired from a section on complex numbers in Art of Problem Solving’s Precalculus book. It was nice to be able to use it to explore a little bit of 2d geometry, too.

# Exploring trig and 2d geometry with 3d printing

This week I’ve been doing a fun 3d printing project with my younger son who is learning trig (from Art of Problem Solving’s Prealgebra book). We have used 3d printing to explore 2d geometry before – see some of the projects here, for example:

3d Printing ideas to explore math with kids

Exploring Annie Perkins’s Cairo Pentagons with kids

Evelyn Lamb’s Pentagons are Everything!

This week I had my son create, code, and then print some simple 2d shapes – the project combines ideas from trig, geometry, and algebra.

Here’s his description of the first shape -> a 3-4-5 triangle:

Here’s the 2nd shape – a 7-6-3 triangle. Creating this shape shows how ideas from introductory trigonometry come into play:

Finally, here’s a regular pentagon that we made yesterday. Unfortunately we made a mistake in the code for the print – mixing up a Sin() and a Cos(), but here is explanation of how to make the shape is correct:

I’d forgotten how useful 3d printing can be as a tool to explore 2d geometry – this week was a happy reminder of how fun those activities can be!

# Sharing Laura Taalman’s slices with my younger son

I had an opportunity to visit Laura Taalman at ICERM today who made me a copy of her latest creation. I couldn’t wait to get home to share it with my younger son:

He had some interesting ideas about what the shape was in the last video. Now I shared where the slices came from and had him explain how Taalman’s creation worked:

This is such a fun way for kids to experience shapes from a different point of view. I’m really excited to see if we can create some similar objects to play with!

# A morning with the permutohedron

Today we are revisiting an old project on a really neat shape -> the permutohedron:

“A fun shape for kids to explore – the permutohedron

I learned about this shape thanks to Allen Knutson at Cornell – he included a fun pic of a large permutohedron in the comment of the blog post above:

He also pointed me to a 3d print on Thingiverse that we used in the last project and again today:

“Permutahedron” by PFF000 on Thingiverse

So, I started today by having the boys describe the 3d printed shape. We have two versions – a larger one that unfortunately broke a little and a smaller – but in one piece! – version. Here’s what the boys had to say about the shapes:

Next I had the boys read the Wikipedia page on the permutohedron for about 10 min and then we discussed some of the ideas that they thought were interesting:

Finally, we built the 2-D permutohedron and showed how it was embedded in a 3d grid:

Definitely a fun project and it is always great to be able to have kids hold interesting math ideas in their hands!

# Exploring introductory trig using 3d printing

My younger son is studying trig right now and I thought it would be fun for him to play around with some 3d curves made with trig functions.

I showed him how to use Mathematica’s ParametricPlot3D[] function and then just let him make some shapes on his own. He settled on a curve that looked like this:

Here’s the code just in case it is not legible in the video:

After he made the curve we printed it – it was really fun to see him working on the print when it was finished. I wish the picture was better!

When everything was finished I asked him to tell me about the curve. I’d not seen the code before and didn’t know there was a stray Cos[x] in it. Talking about that piece of the code led to a great conversation about elementary trig functions (totally by accident!):

Finally, I had him talk about the 3d printed shape:

I really enjoyed this project today – it is fun to use 3d printing to explore so many different areas of math.

# Sharing Vladimir Bulatov’s Tetrahedral Limit Set with kids

Last week I saw a really neat tweet shared by Alex Kontorovich:

I ended up buying Bulatov’s piece from Shapeways and it came today. Here’s a quick video look at it:

When the kids got home from school I asked them to take a look at it and share their thoughts.

Here’s what my younger son had to say about the shape:

Here’s what my older son had to say:

# 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.  Here 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:

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.