@mikeandallie or, easy enough to count 24 square units…0h, yeah, I'm not making the shape circular. But can count 6 square units/quarter pic.twitter.com/WtOMaG62tv

@mikeandallie look! it unfolds completely, so that the centers become the edges and there's that shape you mentioned that's been in hiding. pic.twitter.com/JbCKh6mZi0

Next we compared the shape to the 3d prints it was based on:

After this I showed the boys the tweet above that shows how to unfold the shape into a cube and they were able to recreate the procedure:

Definitely a fun shape to explore – thanks to Paula for sending it to us! It really is amazing how much geometry you can explore just by folding paper ðŸ™‚

We’ve been enjoying going through Kelsey Houstin-Edwards’s new video series. This week’s was a bit more advanced than some of the prior ones, but I gave it a shot with the kids anyway. I tried to focus on connecting the ideas about singularities in the video with some of the 3D printed shapes we’ve been studying from Henry Segerman’s new book.

Also, I’m just getting over a few days with the norovirus, so sorry if this one (including the write up) has a bit less energy than usual.

Here’s the latest PBS Infinite Series video:

Here’s what the boys took away from the video:

Next we looked at a couple of the shapes that Henry Segerman has made to study with shadows. We were able to see (eventually) that the shadow of the north pole would be a point at infinity – or a singularity.

At the end of this video we started looking at a torus, and the conversation took a very interesting topological turn.

So, we landed on a question of what different shapes might be a torus. It took a bit of time to straighten out this idea, but after a few minutes we came to an agreement on what a torus was.

After that we saw that we didn’t have the same singularity problem trying to create a map that we had on the sphere.

After talking about the torus we spent the rest of this video talking about the pseudosphere which has more than one singularity.

So, another great video from Kelsey Houston-Edwards. It was fun connecting her ideas with some of the 3d prints we’ve been studying lately.

Today our math and 3d printing project combined ideas from two great books.Â 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!

We are continuing to explore the different ways for kids to see math with 3d printing. Henry Segerman’s new book has been an incredible resource for us in this long-term project:

Yesterday I asked the kids to pick more shapes from his book to print. My older son picked “Topology Joke” and my younger son picked a shape that we’d already printed, but unfortunately the prior pick didn’t survive an unexpected encounter with a book ðŸ™‚ Here are the shapes and what the kids see in those shapes.

“Topology Joke”

My favorite quote – “A torus somehow equals a coffee cup”

Here’s my younger son looking at the trefoil knot on a torus. The interesting thing to me about his discussion of the shape is that he thought the torus was just as interesting as the knot:

At this point we have close to 50 3d printing projects for kids on the blog. Henry Segerman’s work and Laura Taalman’s work have been incredible inspirational for me. I can’t wait to explore more with how 3d printing can help kids see math in a way that was far more difficult to see previously.

Calculating the volume of 3 intersecting cylinders is a classic calculus problem.Â The 3 cylinder problem caught my attention a few years ago when Patrick Honner shared this video about the 3d printing lab at his high school:

Today we used our 3d printer and the F3 program to explore the intersection of three cylinders. Here’s what they boys had to say when they saw the setup on the F3 program – my older son went first:

Here’s what my younger son had to say:

After the shape finished printing I had the boys talk about their thoughts when they had the shape in front of them. Here’s my older son’s thoughts:

And next my younger son:

Even though this is probably a better Calculus example, I loved being able to share the shape with the boys. It is fun to hear kids talk and wonder about fun shapes like this one.

We’ve done a lot of projects relating to platonic solids and dodecahedrons in particular. A really neat fact about dodecahedrons is that you can use the verticies to put 5 cubes inside!

It isn’t just a mathematical “fun fact” either – the symmetry groups involved play roles in important mathematical theorems.

For today’s project I wanted to explore one cube in a dodecahedron and look at the relationship between the rotations of the cube and the rotations of the dodecahedron.

We started by looking at the dodecahedron by itself:

Next we moved to looking at the cube in the dodecahedron and studied what rotating the dodecahedron did to the cube:

Finally we looked at some 3d printed models that we made to see if these models helped us explore the rotations a bit more:

I was a little disappointed that I made the 3d printed models a bit too small, but I still like how this project went. I’m going to try again with some slightly larger models with my older son.

My older son made a “twisted octahedron” based on one of the examples that comes with the program. When we talked about the shape he wondered how you would calculate the volume of the shape. Today I made some slicing models that you might see in a calculus class to help him see the answer to that question.

I’m still 3 steps below a novice at using F3 but am learning a bit more every day. The program I wrote to make the slicing models is pretty easy (and pretty short) and didn’t take that long to figure out. The code is below – I show it not because it will make sense to everyone, but rather to show how easy it is to make really cool shapes with the F3 program. The code is a very slight modification of F3’s twisted tetrahedron example that caught my son’s eye the other day, and running this code allowed me to make the three sliced examples in the second video below.

So, for our project tonight we first revisited the twisted octahedron and talked about the volume:

Next we looked at the sliced models I made. Unfortunately I made the models way too small. By luck they showed up ok on camera (though sorry for going off screen a few times) and the boys were able to see how the smaller slices converged to the shape we had originally:

I’m not teaching the boys calculus – don’t worry! It still is really neat to be able to show them some of the ideas from calculus using 3d printed models. Can’t wait to play with more shapes this week!

Today we explored the 2-dimensional idea in Kelsey Houston-Edwards’s video in 3-dimensions using 3d printing.

First, though, I asked the boys what they thought about the latest PBS Infinite series video:

Next we took a look at 6 different “spheres” in 3-dimensions which were defined using the different way of measuring distance Kelsey Houston-Edwards introduced in her video. It is really fun to hear kids talking about these shapes, and even more fun to be able to actually hold these shapes in your hand!

The last thing we did was look at the 6 spheres all together. My younger son noticed how much bigger the shapes got as you moved from the L-0.75 norm to the L-4 norm. My older son noticed that some of the shadows of the shapes looked pretty similar even though the shapes didn’t look the same at all:

This was a really fun project to prepare. It is really fun to show kids ideas from advanced math that the wouldn’t likely see in school. It is also really fun to hold these strange shapes in your hand! Lately we’ve been using the F3 program to help us make objects to print – it was a really lucky coincidence to see the new PBS Infinite Series video *after* I learned to use the new program. It took less than 5 minutes to make the .stl files for the 6 shapes ðŸ™‚

We’ve had a blast playing with the F3 program since learning about it a few days ago. The learning curve is pretty steep – at least for me – but holy cow is this program an amazing resource for 3d printing. Here’s a link to our initial project with F3:

Last night both kids and I made some shapes using different aspects of the program. My youngest son (5th grade) made a shape from one of the built in examples. My older son (7th grade) made a shape by modifying one of the examples. Finally, after seeing some of the code in the example that my son used, I was able to build a shape from scratch.

Here’s the discussion of those shapes:

(1) my younger son

(2) my older son

(3) my shape

I’ve become an instant fan of the F3 program and am super excited to keep using to do 3d printing projects with the boys.

I desperately wanted (and still do!) to figure out how to 3d print the last creation in his blog post. Wiggins pointed me to a program by Reza Ali (and sorry for saying “Renza” in the first video) that he thought would do the trick:

I’m grateful to both of their help in pointing me to and helping me understand how this program works. I still don’t quite know how to make the shape from Wiggins’s blog, but part of the reason is that the F3 program is so cool that I’ve been having a ton of fun just playing around with it. (The other part is that I don’t know what I’m doing at all . . . . ðŸ™‚ )

Tonight I showed the program to the boys and played with some “sums” and “differences” of 3d shapes.

Here’s the introduction and what the boys thought a few shapes would look like:

I let the boys pick some shapes that they thought would be fun to see. Here’s what my 5th grader thought would be fun:

We wrapped up with my older son picking a few shapes. He wanted to move some shapes left and right which I’d not done before, but luckily the program was intuitive enough for me to guess how to do it. Small victories . . . . The next thing I’d like to learn how to do is angle the shapes differently so that I can make the “Prince Rupert Cube” shape, which was one of our first ever 3d printing projects: