Comparing Sqrt(x^2 + y^2) and ( Sqrt(x^2) + Sqrt(y^2) )

Last week we used 3d printing to compare $(x + y)^2$ and $x^2 + y^2$:

That project is here:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing

My younger son is still sick today and not able to participate in a math project, so I chose a slightly more algebraically complicated comparison to look at with just my older son -> $\sqrt{x^2 + y^2}$ and $\sqrt{x^2} + \sqrt{y^2}$

Here’s what the shapes look like:

I started the project by reviewing the original project in this series just to remind my son about how we thought about the 3d surfaces in the prior post. He remembered most of the ideas, fortunately, so the introduction was fairly quick.

After the introduction we talked about some basics of the algebra we were going to encounter in this project, namely that $\sqrt{x^2} = |x|$. This part all by itself is a difficult concept to understand and the bulk of the video below was spent talking about it.

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With the difficult part of the algebra behind us we moved on to talking about the surface $z = x^2 + y^2$. What does this surface look like?

I really enjoyed the discussion here – the question is actually a pretty challenging one for a kid to think through.

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Next we tried to figure out what the surface $z = \sqrt{x^2} + \sqrt{y^2}$ would look like.

I think it takes a while to get used to working with graphs of the square root function. My son struggled a bit here to figure out the shape here. Hopefully that struggle helped him

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Now I revealed the shapes and let my son discuss the properties of the shapes now that he could hold them in his hand. There were a few surprises, which was nice 🙂

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I’m really happy about this series of projects. It is fun to explore the variety of ways that 3d printing can help kids explore math.

Learning 3d geometry with Paula Beardell Krieg’s pyramids

Earlier in the week we got a nice surprise when we received a fun little pyramid puzzle from Paula Beardell Krieg:

Our initial project using the shapes is here:

Playing with an amazing present from Paula Beardell Krieg

I thought a follow up project would be fun, so I decided to try out a basic exploration in 3d geometry. The goal was to make these shapes ourselves using Mathematica, then to 3d print them, and finally to play with the new shapes to see that they were indeed the same.

We started by talking about the shapes in general and see if we could identify some very specific properties of the shape using coordinate geometry:

Next we talked about how to describe the planes that formed the boundary of the shape. It was fun hearing my 5th grader try to figure out how to describe the planes (and regions) we were studying here. One other challenge here is that we were also trying to describe the 3d regions above and below these planes.

Now came the special challenge of finding a mathematical way to describe the hard to describe plane in the shape. I had to guide the discussion a bit more than I usually do here, but the topic of finding the equation for a plane is pretty advanced and something that kids have not seen before.

Having written down the equations, we went up to look at the Mathematica code I’d used to make the shapes. The boys were able to see that the first shape had exactly the same equations we’d written down, and they were able to see that the equations for the 2nd shape were not any more difficult.

The shapes printed overnight and we had an opportunity to play with them this morning. It is pretty neat to hear them compare the shapes and see that, indeed, the shapes we made are really the same as the shapes Paula sent us.

So, there’s quite a lot we can study with Paula’s shapes. You’ve got the potential to study folding patters, basic 3d geometry, the volume formula for a pyramid, and even 3d printing! Fun how such a seeming simple idea can lead you in so many different directions.

A fun shape for kids to explore: the Permutohedron

I learned about permutohedrons from a comment by Allen Knutson on a prior blog post. See the first comment here:

A morning with the icosidodecahedron thanks to F3

I prnted the shape from Thingiverse and it was amazing!

“Permutahedron” by PFF000 on Thingiverse

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 🙂

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.

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!

A morning with the icosidodecahedron thanks to F3

A few weeks a go I saw this shape in a display case at the MIT math department:

The shape is mislabled, unfortunately, it is an icosidodecahedron. We’ve already done a few projects based on the shape. Last week’s project is here:

A zometool follow up to our Cuboctohedron Project

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

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

(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!

A Zometool follow up to our cuboctohedron project

Earlier in the week we studied the cuboctahedron:

That project is here:

Playing wiht the Cuboctahedron

Also earlier in the week I saw these shapes displayed in the MIT math department:

The chance encounters with these shapes this week gave me the idea to revisit them today and see if we could build them with our zometool set. The second shape, I think, is mislabled in the MIT display case – or maybe they are just using a less common name. The usual name is the icosidodecahedron, and it is also a shape we’ve seen before:

I started the project today by showing the shapes to the boys and asking what they knew about them:

Then we went to the living room to build the shapes. The only tricky part is that the cuboctahedron needs green struts. As always, the wonderful thing about the Zometool set is that you can go from seeing these shapes on a page to holding them in your hand almost immediately!

The last part of the project was building the dual shape of the cuboctahedron. I wasn’t sure if the zome set would let us do this since you can’t exactly find the center of the triangles with zome – but we did catch a lucky break! The dual is also a shape we’ve seen before 🙂

This project was really fun – exploring geometry with our Zometool set is one of my favorite activities!

Playing with the Cubeoctahedron

Last night I was flipping through the book I bought to understand a bit more about folding – Geometric Folding Algorithms by Erik Demaine and Joseph O’Rourke:

and I ran across a short note on the cuboctahedron. The boys were taking a short trip today (school vacation week!) and I was looking for a short project to do before they left – folding up the cuboctahedron seemed perfect.

Making my life much easier was a template on Wolfram’s website:

Wolfram’s folding template for a cuboctahedron

Here’s what the boys had to say after creating the shape:

After the short discussion about the shape we went upstairs to look at the shape using the F3 program. My idea for the ~10 min discussion here was inspired by a talk by Keith Devlin I saw over the weekend:

I thought that an approach similar to a game with our F3 program would help the boys create the shape.

Here’s how we got started. The F3 program allows us to create a cube and an octahedron. It also allows you to add and subtract shapes. How can we use these 4 ideas to create the cuboctahderon?

I think the video here really shows what Devlin calls “mathematical thinking.” The conversation here was really fun (for me at least!) since trying to discuss the ideas through equations would be impossible. However, the geometric ideas are accessible to the boys via the F3 program, just as the number theory ideas are accessible to kids through Devlin’s “Wuzzit Trouble” program.

I broke the discussion into two pieces – at the start of the 2nd half of the discussion we are trying to figure out how to – essentially – flip the shape inside out. My son comes up with an idea that was very different than what I was expecting, and it worked 🙂

Revisiting James Tanton’s Tetrahedron problem

A little over 2.5 years ago I saw this very neat question from James Tanton:

The question led to a really fun – and also one of our first – 3d printing projects:

James Tanton’s geometry problem and 3d printing

We’ve now got a few more years of 3d printing under our belts and a new program we are using – F3 by Reza Ali – is opening completely new 3d printing ideas for us.

Somewhat incredibly, F3 has a one line command that draws all of the points that are a fixed distance away from a cube. Here’s that beautiful shape:

Seeing that command inspired me to revisit James Tanton’s old question. I wasn’t quite able to do it in one line (ha ha – my programming skills are measured in micro-Reza Alis . . . .), but I was still able to make the shape. Here’s how it looked on the screen:

After the boys got home from school we revisited the old project together and used both the old and the new 3d prints to help us describe the shape (sorry for the noise in the background – that’s a humidifier I forgot to turn off):

Maybe because it is one of our first projects ever(!), but I love this problem as an example of how 3d printing gives younger kids access to more complex problems.

Learning math by studying 3d printing

My son spent the last couple of months preparing for the AMC 10. Now that the test is behind him I’m going to spend some time with him studying 3d printing.

Today we looked at some simple code in the F3 program:

The details of the code don’t matter that munch – all the code is doing is testing whether or not a point is inside of a sphere by checking whether or not the distance from that point to the center is greater than or less than the radius.

Immediately two ideas come to mind:

(i) how do we compute distance in 3 dimensions?

(ii) is that distance measure unique.

So, after 1 minute of looking at code we went to the whiteboard 🙂

Our previous 3d prints of the sphere and torus in different L^p metrics were still on the table, so I used those as props.

The first topic was distance in two dimensions:

The second topic was distance in three dimensions:

The last topic was how the L^p metrics vary as p varies – it was lucky we had the spheres handy 🙂

Today’s conversation was actually a nice surprise – I think there’s going to be quite a lot of fun math review that comes from studying 3d printing more carefully.

Sharing advanced ideas in math with kids via 3d printing

Yesterday (after a few false starts!) I printed several different versions of the torus in different L^p metrics. Here they are next to spheres in the corresponding metric

The idea was inspired by an old project that was inspired by a Kelsey Houston-Edwards video

Sharing Kelsey Houston-Edwards’s Pi video with kids

Prior to the prints finishing I talked through some of the shapes as they appeared on the computer with my younger son:

Exploring different L^p versions of the torus

When the various torus prints were done I asked each of the boys to tell me what they thought about the shapes. I love how 3d printing allows you to share advanced ideas about math with kids so easily!

Here’s what my younger son had to say:

Here’s what my younger son had to say:

These are the kinds of math conversations that I’d like to have with kids.

Our second facets project

We did this about a week ago and I never got around to publishing it. Both kids are sick today so it seemed like a good day to revisit the old movies.

Our first projects with our Facet set is here:

Our Facets have arrived!

The kids have enjoyed making little creations with the Facets ever since.

Here’s how their 2nd projects looked – my younger son went first. It is so fun to hear a week later how many different topics we covered in the talk about the ring he made:

Here’s my older son’s work. He made an interesting shape that we tried to extend with the camera off. Unfortunately that shape was too heavy for the magnets and collapsed. We got a little lucky, though, and some interesting shapes survived the collapse and we turned them into new, fun shapes:

I’m really happy with the Facets and can’t wait to do more projects with them!