Finding the coordinates of the vertices of a Tetrahedron and an Octahedron

My younger son is starting to learn about coordinates in 3 dimensions. I thought that spending a little time finding the coordinates of the corners of a tetrahedron and an octahedron would make for a nice project this morning.

We started with the tetrahedron and found the coordinates for the bottom face. Once nice thing about the discussion here was talking about the various choices we had for how to look at the tetrahedron:

Having found the coordinates for the bottom face, we now moved on to finding the coordinates for the top vertex:

Now we moved on to trying to find the coordinates for the corners of the octahedron. Here the choices for how to orient the object are a little more difficult:

Finally, we talked through how we would find the coordinates of the octahedron if we had it oriented in a different way. This was a good discussion, but was also something that confused the boys a bit more than I thought. We spent about 10 min after the project talking through how to find the height. Hopefully the discussion here helps show why this problem is a pretty difficult one for kids:

A Zometool Icosahedron project inspired by Steve Phelps

I saw a neat tweet from Steve Phelps yesterday:

It looked like it could make a neat project both on the computer and with our Zometool set.

First I had my younger son look at Phelps’s visualization – one really interesting observation he had was that the intersecting lines inside the icosahedron dodecahedron:

Nest I had my older son look at a similar program in Wolfram’s Demonstration Project. The thing that caught his attention was all of the underlying structure:

We also created a zometool version of the icosahedron with all of the diagonals. We tried to see if we could see the same interesting things that we saw in the computer programs using the Zome shape:

Later in the day we did build a slightly larger icosahedron in which the diagonals did intersect on a zome ball. This allows you to see the dodecahedron that my younger son thought was there:

Visualizing the 5d permutohedron with kids

Last night as part of a linear algebra project I was doing with my older son, we found out that you can orient a 4d permutohedron in 3 space so that all of the vertices have integer coordinates:

Today I wanted to explore that idea a bit more and also include my younger son. So, I thought it would be fun to see if we could find a way to see what the 5d permutohedron looks like by looking at slices of it in 4d.

I started by reviewing the 3d permutohedron and how it is embedded in 2 dimensions. It was nice to go back to the beginning here – especially so that we could explore how slicing with lower dimensional slices works.

Next we tried the same “visualization by slicing” idea with our 4d permutohedron embedded in 3 dimensions:

Finally, and sorry this one is long, we got to the heart of today’s project. Here we’ll be using some code I wrote in Mathematica to view 3d slices of the 5d permutohedron emedded in 4d space. It is close to a miracle that I was able to get these visualizations to work correctly – maybe the extra hour this morning helped! It was super fun to hear the boys talk about what they saw with these shapes:

Labeling each vertex of a permutahedron is a terrific mathematical exercise for kids

Yesterday we did a fun project exploring a permutohedron:

A Morning with the permutohedron

Last night I thought it might be neat to have the kids try to label the vertices of a permutohedron with the permutations represented by each vertex. Fortunately, it was possible to build a truncated octahedron with the green Zometool struts:

We started out today’s project by talking about the rules for making a permutohedron in different dimensions. Here I used the labeling of the permutation of 3 objects as a base case to make sure the boys understood the directions properly.

Next I had the boys label each vertex of the permutahedron with the permutation of {1,2,3,4} that the vertex represented. Then, they talked about the process of figuring out the right labels.

I’m sorry that the video below runs 10 min, but if you listen to the whole discussion I think you’ll see that seemingly straightforward act of labeling these vertexes is a terrific mathematical exercise for kids.

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:

permutohedron

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!

A neat project with a dodecahedron

Saw a really neat tweet this morning:

I thought it would be fun to see what the boys thought of this shape and then try to building using our Zometool set.

First I showed them the video:

Next we spent 20 min building the outside shell of the shape, but for now left the inside mostly empty. Here’s what the boys thought of the shape:

Finally, here is the completed shape – it is a nice little miracle that we could make the whole thing with the Zometool set!

Such a fun project! Happy for the lucky break from twitter this morning 🙂

A bonus project on a Zometool icosahedron

We’ve done two projects on platonic solids recently:

Talking about Angles in Platonic Solids

Following up on our angles in platonic solids project

In the last project my younger son explored two different kinds of “golden rectangles” inside of the icosahedron. I thought it would be fun to try to fill in the entire shape with the rectangles, so today the boys took on that challenge.

Here’s their discussion of the shape made by filling in all of the large golden rectangles in the icosahedron:

Next we turned to the shape made by filling in the smaller golden rectangles. These were a little harder to make. Since the first shape took a bit longer to make than we expected, we only filled in 10 of these rectangles and avoided the problem of dealing with ones that overlapped.

To wrap up we removed the struts from the original icosahedron to get a better view of the shape formed by the rectangles:

Definitely a fun project. As always, it is incredible how easy (and fun!) it is to explore 3d shapes with a Zometool set.