Tag zometool

Extending Vsauce’s 4 dimensional shadow tweet a bit

Saw a fun tweet from Vsauce before we left for vacation:

He later shared the shape, too:

Using shadows is a an incredibly fun way to explore complex shapes. Henry Segerman gave an amazing talk about the idea last fall at MIT:


I think playing with these sorts of shadows is a great way to share complex shapes with kids, too (we also used a Zometool set!):

Playing with shadows inspired by Henry Segerman

4-dimensional Shadows

Playing with our Zometool model of Bathsheba Grossman’s “Hypercube B” was especially cool – you can see some of the same effects as in the Vsauce video, though I think the two shapes are a little different.

Though this project shows that there are a few different Zome versions of the shape, so maybe my blue-strut version isn’t all that different from Vsauce’s 4-d cube after all:

Bathsheba Grossman’s “Hypercube-B” part 2

Anyway, as Vsauce’s tweet shows, exploring shapes via shadows is fascinating. It is also a really fun way to introduce kids to shapes that they’ve probably never seen before!

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!

Taking about Kate Nowak’s shape

Saw this neat drawing from Kate Nowak the other night:

I was interested to see if we could make the shape from our Zometool set, and . . . .

The boys really enjoyed making the shape last night and both also made several comments about how interesting it was. This morning we talked about it a bit. Both kids focused on symmetry. I spent a bit more time with my older son exploring the different kinds of symmetry, but it was great to hear what both kids had to say. It really is an amazing shape!

Younger son first:

Older son next:

This was a really fun project. The shape didn’t take that long to build, which was lucky. It is always fun to be able to pull out the Zome set to explore something that we saw on Twitter 🙂

Using 3d printing to talk symmetry with kids

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.

A project inspired by an AMC 12 octagon problem

The problem pictured below from the 2003 AMC 12 gave my son some trouble:

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We talked through it together a few days ago, but I thought it would be fun to try to do an octagon-inspired math project today.

We started with the problem and then talked a bit about a 3d print we found on Thingiverse:

Next we took a look at a version of the 3d printed shape that we made from our Zometool set. You can’t make a regular octagon with a Zometool set, and the fact that our shape didn’t have a regular octagon led to a good discussion:

For the last part of the project we tried to find the volume of our truncated cube.

A fun coincidence with an Eduardo Viruena creation

I got some great feedback from Eduardo Viruena on the project we did with one of his math designs:

A short project inspired by a Holly Krieger tweet

One of his other designs he pointed me to was this one:

A small stellated dodecahedron approximated by dodecahedra

Here’s his picture:


I printed it over the course of the day (took about 6 hours) and showed it to my younger son when he got home from school. Here’s he described the shape, including noticing one very interesting pattern that he thought would form an Archimedean solid:

It turns out that the shape he saw would indeed be an Archimedean solid. In fact, it the exact solid we did a project on a few weeks ago!

Here’s that project:

Revisiting our Zometool Snowman

Which was inspired by this tweet from Eli Luberoff:

The Snowman is still up in our living room (which I’ll attribute half to coincidence and half to laziness . . . . ) so we looked carefully at the two shapes:

Amazing what kids notice when they look at mathematical objects!

Revisiting our Zometool Snowman

When we first moved into our house we did a couple of fun and large Zometool projects because we didn’t have any furniture 🙂

This week I saw a fun tweet from Eli Lubroff that reminded me of one of those projects:

Here’s a part of that old project 🙂


Today we revisited that old snowman and had the boys talk about each of the Archimedean solids in the shape. This is a fun project – not just because the shapes themselves are cool – but you get a nice opportunity to talk about counting and symmetry. You’ll see in the videos that my older son is a bit more comfortable with the idea, but my younger son seems to catch on by the 3rd video.

Here’s a link to all of the Archimedean solids on Wikipedia:

The Archimedean Solid page on Wikipedia

And here’s our project:

First the bottom of the snowman – the Truncated Icosidodecahedron

Next was the Rhombicosidodecahedron

Next was the Icosidodecahedron

Finally the Archimedean Solid Snowman 🙂 Two years later and he still fits!

Definitely one of my all time favorites and a really fun way to discuss counting and symmetry!

The Koch Snowflake with Squares

I asked the boys what they wanted to talk about today and got a fun response – the Koch Snowflake with squares!

Luckily we still had the Zome set out from the “tribones” project from last week – so making the first couple of iterations wasn’t that hard.

Before we started, though, I asked the boys what they thought the shape would look like:

While they were building I searched for something on line that would let us play with this particular fractal. I found these two Wolfram Demonstration projects:

Creat Alternative Koch Snowflakes by Tammo Jan Dijkema


Square Koch Fractal Curves by Robert Dickau

We’ll explore these to demonstrations below.

Here’s what the boys had to say about the first three iterations of the square Koch Snowflake. A fun thing that happened here was that during the discussion the boys found a small mistake in the construction of their level 3 curve:

Next we moved to the computer to explore the Wolfram Demonstration projects. First up was Robert Dickau’s “Square Koch Fractal Curves.” Sorry that this video (and probably the next one, too) is so fuzzy – it looked ok in the view finder. Oh well.

Finally, we explored Tammo Jan Dijkema’s “Create Alternative Koch Snowflakes” demonstration project. This project allows you to alter the fractal. The boys had a lot of fun playing with this project. Again, sorry for the fuzz in the video.

Math school year in review part 1 – our tiling projects

Roughly a year ago the boys started their first year in school after 5 years of home schooling. Both kid’s packets for the next school year arrived last week. That got me daydreaming about the math we’ve done in the last year. Turns out that we’ve done a lot and I wanted to write about some of it before it all slips out of my mind.

One subject that we’ve spent a fair amount of time on this year is tilings. Not by design but I just happened to see a lot of neat tiling ideas in the last year.  Here’s a review of the tiling projects we did in the last 12 months:

(1) The project from Zome Geometry that got us going

This project was one of the few ones that I didn’t film.  The reason was that we had several kids from the neighborhood over working on it and I don’t feel comfortable filming kids that aren’t mine!

Anyway, this was a really fun project from Zome Geometry by George Hart and Henri Picciotto

Zome Tilings


(2) That project led to two 3d versions:

The natural thing to do after this project was to look at ways that you could cover 3 dimensional space with shapes – again we made use of our Zometool set:

Tiling 3-dimensional space with our Zometool set


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(3) A problem from a UK math exam led to a fun tiling project

I saw this neat problem from a UK math test circulating on Twitter back in February.

The UK Intermediate mathematics challenge part 2

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(4) A domino counting exercise form Jim Propp

We’d done a couple of projects based on Jim Propp’s blog, he thought that we might enjoy studying how 2×1 dominoes tile a 2xN square.  The project was so fun that we actually did it twice!

A fun counting exercise for kids suggested by Jim Propp

Counting 2xN domino tilings

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(5) Propp’s suggestion above came after we did these two projects on the Arctic Circle Theorem

I learned about the Arctic Circle theorem from a graduate student at MIT who thought it might be possible to share this fairly advanced mathematical idea with kids:

The Arctic Circle Theorem

A second example from tiling the Aztec diamond

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(6) Dan Anderson’s Gosper Curves

Maybe stretching a little to call this tilining, but we had fun exploring how the Gosper Island’s that Dan Anderson sent us fit together:

Dan Anderson’s Gosper curves

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(7) Inspiration from Eugenia Cheng’s Shapes video

I saw this neat video from Eugenia Cheng over the summer:

Thinking about how to use it with my kids inspired these two projects:

Tiling pentagon cookies

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

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(8) Richard Stanly’s Tiling presentation

Finally, just last week I stumbled on a presentation that Richard Stanley – a math professor at MIT who specializes in combinatorics – had put together about tilings.  There were a couple of ideas that were accessible to kids:

Talking through some examples from Richard Stanley’s tiling presentation

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Talking through some examples from Richard Stanley’s tiling presentation

I stumbled on this incredible presentation about tilings from Richard Stanley yesterday:

Richard Stanley’s presentation on Tilings

We’ve actually done many projects on tilings in the past if you search the blog – some of the fairly recent ones are:

Counting 2xN domino tilings

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Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

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Zome Tilings


Today we looked at two fun examples from Stanley’s paper -> tiling a chess board with dominoes and tiling a hexagon grid formed by the triangular numbers with “tribones.”

First up with the chess board. The problem here is pretty famous and a really fun one to try out with kids. Just in case you’ve not seen it before and want to try it out yourself, the problem is: If you remove two opposite corners of a chess board, can you tile the remaining shape with 2×1 dominoes:

Next we discussed the problem my younger son asked about – what happens if you removed two random squares of opposite color?

Now we moved on to the tribones and the hexagon grid. Here’s a quick discussion / introduction to the problem from Stanley’s paper:

Next I intended to have them try to build the T(9) shape from the tribones, but we took a little detour first to try to figure out why building T(6) from tribones was impossible. It probably took 10 minutes for the kids to find the argument, but it is was fun work. I wish I had left the camera running for it, but I didn’t. Here’s a short summary of the argument:

Finally, we wrapped up the project by trying to construct T(9) from the tribones:

So, a really fun weekend of tiling. I’m really happy that I stumbled on Stanley’s presentation yesterday!