With 4 of them on the table I asked each of the boys what they thought the shapes were and then let them play around with them. After they played for a bit I put a book on the shapes and asked them how they thought the book would move as the shapes rolled.

Here’s what my younger son thought:

Here’s what my older son thought:

I always find it fun to hear what kids think about complicated shapes. Lots of neat ideas and then a good “wow” when you learn the secret property!

We started the project today by examining the shape and comparing it to a few other shapes we printed. The comparison wasn’t planned – the other shapes just happened to still be on the table from prior projects . . . only at our house ðŸ™‚

Next we talked about permutations and the basic idea we were going to use to make the permutohedrons. We drew the 1 dimensional version on the whiteboard and talked about what we thought the 2 dimensional version would look like.

We used our zometool set to make a grid to make the 2 dimensional permutohedron. Lots of different mathematical ideas for kids in this part of the project -> coordinate geometry, permutations, and regular old 2d geometry!

Next we went back to talk about how PFF000’s shape was made. Here’s the description on Thingiverse in case I messed up the description in the video:

“The boundary and internal edges of a 3D permutahedron.

The 4! vertices are given by the permutations of [1, 3, 4.2, 7], with an edge connecting two vertices if they agree in two of the four coordinates. The 4D vertices live in a 3D hyperplane, namely the sum of the coordinates is 15.2.
Designed with OpenSCAD.”

This part of the project was a little longer, but worth the time as both the simple counting ideas on the shape and the combinatorial ideas in the connection rules are important ideas:

Finally we wrapped up by taking a 2nd look at the shape and also comparing it to Bathsheba Grossman’s “Hypercube B” which was also still laying around on our project table!

This was a really fun project that brought in many ideas from different areas of math. I’m grateful to Allen Knutson for the tip on this one!

Finding the volume of the intersection of two cylinders is a common calculus problem. The shape also plays a role in this old (for the internet!) video from Brooklyn tech that inspired me to get a 3d printer:

Today for a fun project to start the week I decided to share the shape with the boys and see what they thought about it. My younger son went first:

After playing on the computer I had him explore the printed version of the shape – make sure to stay to about 1:25 to hear where he thinks this shape might occur in “real life” ðŸ™‚

Next my older son played with the shape on the computer. He remembered seeing it before in a project from a month ago on the intersection of 3 cylinders:

Next he played with printed shape. I asked him to describe how he thought you’d be able to figure out that the shape was made out of squares – I thought his answer was pretty interesting. This question gets to the math ideas behind the calculus problem.

It is sort of fun for kids to see and play with shapes like this – no need to wait for calculus anymore to explore interesting shapes!

And there’s our fun Zometool Snowman, too, where the icosidodecahedron is the head:

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The name suggests that it is made from a combination of an icosahedron and a dodecahedron – but how?

Ahead of the project I made a few shapes:

(i) the icosidodecahedron
(ii) a dodecahedron with an icosahedron removed
(iii) an icosahedron with a dodecahedron removed

Here’s what the boys thought of those shapes:

Next we went upstairs to play with some code in the F3 program. Looking at the video now I see that I forgot to publish it hi def – sorry about that. I hope our explanation of the code is good enough if the code is too fuzzy to read:

Definitely a fun little project – it is so fun to be able to play with these shapes on the computer and then hold them in your hand!

As a follow up to our last two projects with 3d printed triangles, I thought it would be fun to try out a similar project. The point of the 2nd project wasn’t the geometry, though, it was to use the process of making the shapes as a way to review some basic ideas about lines.

So, I started by showing my older son the basic idea – we wanted to write down the equations of the three lines that border a 5-12-13 triangle. Since we have a right triangle the equations aren’t too difficult, but are still useful for a simple review:

Next we went upstairs to Mathematica to make our 3d template using the function RegionPlot3d:

After the prints finished we played around with them a bit to see which triangles we could make with the same area:

I think that making little shapes like this might be one of the best educational uses of 3d printing. Kids get an opportunity to apply some basic math knowledge and create some fun shapes!

During the day I was just playing around with the triangles and found a couple of other fun ideas from geometry to show the kids.

First, an alternate proof that the two shapes have the same area. Almost a proof without words! It comes from the fact that the two shapes can come together to form a 6-8-10 triangle:

The next geometric idea involved a right triangle being inscribed in a circle:

It was a really nice surprise that there were other fun (and important!) geometric ideas that our shapes could be used for. We’ve used 3d printing before to play with 2d geometry – including triangles:

I’ve been binge listening to math podcasts lately. In one – and, sadly, I don’t remember which one! – I heard a neat problem about triangles. Last night Paula Beardell Krieg shared Suzanne von Oy’s blog post which reminded me of the problem:

After seeing von Oy’s post I threw together some shapes from the puzzle and set them to print overnight. This morning I went through the puzzle with both kids.

Here’s the puzzle: You have triangles with sides 5-5-6 and 5-5-8. Which one has the larger area?

My older son went first:

My younger son went second – my older son has much more experience with geometry so I thought having them work separately would be better.

I think this is a great puzzle / problem for kids learning geometry.

We’ve been doing a little bit of work with knots lately. Today we were studying the knot with 5 crossings, and it wasn’t so easy.

I’d guess ahead of time that 5 crossing would be tricky. There’s a lot to keep track of! Even what seems like a simple task – making a knot with 5 crossings out of rope – isn’t so easy. See if you can spot the problem:

The boys didn’t notice the problem with their knot, yet, but the problem quickly became clear when they started playing with it:

So, we started over . . .

Having now made a knot with 5 crossings, we ended the project by trying to determine which of the two knots with 5 crossings that it was. We got a little bit of a surprise when it turned out that we’d actually made the mirror image of one of the knots we printed. That was an accidental good lesson, though – mathematicians consider those two knots to be the same even though they are not always the same.

Today we moved on to the knots with 4 and 5 crossings. We started off by comparing the trefoil knot with the 4 crossing knot – what is the same? what is different?

Also – the 3 white knots (1 with 4 crossings and 2 with 5) that appear in this projects come from Mathematica’s knot data collection and the red knot (the trefoil knot) was designed by Laura Taalman.

Next I had the boys try to make a knot with 4 crossings out of a rope. It is not as simple as it seems! One nice thing about making these knots out of rope is that they also begin to discover some of the ways you can have crossings that can be undone.

Now we compared the knot we made from the rope to the 3d printed knot with 4 crossings. In the last project we had quite a lot of difficulty comparing the different versions of the trefoil knot. Here, though, comparing the knot in our rope to the 3d printed knot was not too difficult.

Finally, we wrapped up this project by inspecting and comparing the two knots with 5 crossings. It is very interesting to hear what kids see in these two knots, and also fun to hear their ideas for how you might determine that these two knots are actually different.

Definitely a fun project. I really like exploring these knots with the kids. It makes me wonder if there is a way to go to the next level and help them understand some of the ideas that help you understand when two knots are different?

I’ve been doing a lot of thinking and playing with our various 3d printed knots lately. It feels like there are lots of great projects for kids here, but I’m struggling a little to find them.

Last night I tried something pretty simple – take several different versions of the trefoil knot and have the kids try to recreate those versions with a knotted rope.

Here are the shows and some initial thoughts about them from the boys. These knots were designed by Henry Segerman and Laura Taalman:

Next we started trying to make the shapes – first the “easy” ones ðŸ™‚

Next we moved on to some more difficult shapes – in particular the 2nd one gave the boys quite a bit of difficulty. Making the connection between these two versions of the trefoil knot isn’t completely straightforward:

I let the boys try to finish making the new knot shape with the camera off. It took a few more minutes. In this video they show how to go back and forth between the two versions:

So, definitely a fun project, but a little more difficult than I expected. We’ll see how difficult playing with the knot with 4 crossings is tomorrow.