Tag: 3d geometry

Sharing a new “picture” of the Milky Way with kids

I saw a new image of the Milky Way last week thanks to these two tweets:

After seeing the first tweet from Kayley Brauer I was hoping to find a way to talk about this new result with the boys, but didn’t really know what to do. Thanks to the tweet from Dr. Chanda Prescod-Weinstein, I learned that the LA Times had put together a terrific presentation that was accessible to kids.

First, I had my older son read the article on his own and then we talked through some of the ideas he had after reading it.

I thought that reading the article on his own would be a little too difficult for my younger son (he’s about to start 8th grade) so instead of having him read it on his own, we went through it together:

Obviously I’m not within 1 billion miles of being an expert on anything related to this new image of the Milky Way, but it was still really fun to talk about it with the boys. I’m very happy that advanced science projects like this one are being shared in ways that kids can see and experience.

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.

Following up on our “angles in Platonic solids” project

Yesterday we did a fun project on angles in Platonic solids:

Talking about Angles in Platonic Solids

We ended up getting a really neat comment from Allen Knutson on that project. He said:

“You should look for the three orthogonal golden rectangles in an icosahedron! They’re easy to see in a Skwish toy.”

My older son was working on a different math project today, so I had my younger son build an icosahedron out of zome and look for those rectangles. Here’s what he had to say after building the shape:

During his description he found a second rectangle. So, off camera, he filled in that rectangle and then had a bit more to say:

So, thanks to Allen Knutson for the comment that inspired this project, and thanks (as always!) to Zometool for making it so easy to get kids talking about math!

Finding the volume of a rhombic dodecahedron with our zometool set

Yesterday we did a neat project inspired by a tweet from Alex Kontorovich:

Sharing a 3d geometry idea from Alex Kontorovich with kids via zometool

At the end of that project a question about finding the volume of a rhombic dodecahedron came up. Since I was going to be out this morning (and my older son was working on a calculus project) I asked my younger son to play around with the Zometool set and see if he could actually find the volume.

Fortunately he was able to – here’s how he described his work:

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Sharing a 3d geometry idea from Alex Kontorovich with kids via Zometool

I saw an interesting tweet from Alex Kontorovich earlier this week:

We’ve looked at but the Cuboctahedron and the Rhombic dodecahedron before, but I thought it would be fun to revisit the shapes. I also hoped that we’d be able to recreate the shape in the picture with our Zometool set.

So, first we built a cuboctahedron and the boys talked about what they saw in the shape:

At the end of the last video the boys thought that the dual of the cuboctahedron would possibly also be another cuboctahedron. Off camera we built the dual, and happily were able to recreate the shape from Kontorovich’s shape!

They were a little worried that we didn’t have the “true” dual, but I think they came around to believing that these two shapes were indeed duals:

Definitely a fun project – it is always fun to see what you can make with a Zometool set. Maybe tomorrow we’ll revisit an old project of finding the volume of a rhombic dodecahedron. That’s another project which Zometool really brings a lot to the table.

A fun calculus problem -> folding a circle wedge into a cone

I’m a few days late publishing this exercise – my son finished up the section on applied max / min problems last week. But I thought his work on this problem was fascinating and wanted to publish it even if it was a little late.

So, last week my son came across this max / min problem in his calculus book:

Screen Shot 2018-10-01 at 4.24.19 PM

It gave him a little trouble and since I was on the road for work it wasn’t so easy to help him. We went through the problem when I got back from a trip -> I thought it would be fun to start from the beginning and actually make some cones before diving into the problem.

Next we started down the path of trying to work through the problem. Here’s how he got started:

In the last video he was able to write down an expression for the volume of the cone in terms of the angle of the wedge. In this video he writes down a variant of that expression (the square of the volume) and gets ready to find the maximum volume:

Now that he has a relatively simple expression for the volume squared, he finds the derivative to find the angle giving the maximum volume:

Finally – he calculated the maximum volume. The expression for the angle is a little messy, but the maximum volume has a (slightly) easier form.

Overall, I think this is a great problem for kids learning calculus. It also pulls in a little 3d geometry and 2d geometry review, which was nice.

With this section about applied max / min problems done, we are moving on to integration 🙂

Playing with Archimedean solids

For today’s math project we are doing a 2nd project from George Hart and Henri Picciotto’s Zome Geometry:

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I asked the boys to pick three shapes from the section on Archimedean solids. Here’s what they picked:

Shape 1: A Truncated Icosahedron

You start with a triple length icosahedron:

They you truncate it:

Shape 2: A truncated dodecahedron

Start with a dodecahedron with sides made from two short blue struts and 1 medium blue strut:

Now truncate it:

Shape 3:

Truncated Octahedron;

Start with an octahedron with side lengths equal to 3 long green struts.

Now truncate:

Two projects from Zome Geometry

For today’s math project I asked the boys to pick a project from George Hart and Henri Picciotto’s Zome Geometry:

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My younger son picked a project about “Rhombic Zonohedra” which led to a terrific discussion about quadrilaterals and 3d geometry:

Ny older son picked a project on stellations of a dodecahedron. He was a little confused by the directions, but sorting out the confusion led to a great discussion.

I wish every kid everywhere could have the chance to play around with a zometool set.

Revisiting folding a dodecahedron into a cube

Folding a dodecahedron into a cube has been one of my favorite projects to do with the boys. Our first few projects about a “dodecahedron folding into a cube” are here:

dodecahedron fold

A neat post from Simon Gregg showing that a dodecahedron can fold into a cube

Can you believe that a dodecahderon folds into a cube?

(see the link above for the source of the amazing GIF on the right of the screen!)

Some 3D Geometry for Pamela Rawson

Today I had the boys work through the whole project on their own – just stopping every now and then to check in and hear about the progress.

Here are their initial thoughts after building the dodecahedron:

In the second part of the project the boys constructed one of the cubes that can be inscribed in a dodecahedron:

For the 3rd part of the project they “folded” the dodecahedron into a cube

Finally, the boys connected up the zome balls inside the cube and found an icosahedron.

Folding up the dodecahedron into a cube is one of my all time favorite math projects.  It is such a surprise that the two shapes can be connected in this way, and it is really fun to explore this connection with our zometool set!