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:

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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!

 

Playing with Laura Taalman’s 3d printable “Scutoid”

Saw a great tweet from Laura Taalman over the weekend”

That shape was just “discovered” and is discussed on this New Scientist article:

oops – that tweet gives me a good picture, but the article itself is behind a paywall.  Here are two free articles:

Gizmodo’s article on the Scutoid

The article in Nature introducing the shape

Last night I had the boys play with the shape (and I did not tell them what it was).

Here’s what my older son thought about it – sorry that it is a little hard to see the shape in the beginning. I add more light around 1:00 in:

Here’s what my younger son thought:

I thought it was interesting to hear that both boys thought that this shape would not appear in nature. I’ll have them