Tag 3d geometry

Alexander Bogmolny shared one of my all time favorite problems this morning

This tweet brought a big smile to my face this morning:

This is an absolute treasure of a 3d geometry problem, so if you’ve not seen it before definitely take some time to ponder it.

I asked my younger son to play around with the problem using our Zometool set. Here’s what he found:

I love that the Zometool set helps make this problem accessible to kids.

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Revisiting Joel David Hamkins’s “Graph Theory for Kids”

A few years ago we did a fun project with Joel David Hamkins’s “Graph Theory for Kids”:

Going through Joel David Hamkins’s “Graph Theory for Kids”

Here’s the link to Hamkins’s notes for the project:

Graph Theory for Kids

This project was also inspired by the project we did yesterday on the graph isomorphism theorem:

Sharing Lazlo Babi’s graph isomorphis talk with kids

For today’s project I printed two copies of Hamkins’s booklet and had the boys work through it on their own. After they were finished, we talked through the project after they were finished. Here’s the conversation broken into 4 parts – as you’ll see, Hamkins has made an absolutely fantastic project for kids:

Part 1: An introduction to graphs and one surprising property

Part 2: Looking at some more complicated or “extreme” examples and also illustrating how some of the more complicated graphs make for nice counting exercises for kids

Part 3: Now a few examples that the kids made on their own – this part led to a nice discussion about crossings

Part 4: Some 3d shapes and a really fun observation from my older son about the sphere

Counting paths

This week both kids had a homework problem in their enrichment math program that involved counting different paths on the edges of a cube.

I thought it would be fun to use those problems as a way to visit the ideas of counting paths in a lattice.

I started the project with a pretty standard path counting problem -> counting the number of paths that go from corner to corner in a rectangular lattice:

Nice I changed the shape of the lattice and ask the boys how they thought the number of paths would change:

Now we moved on to a problem that was similar to the problem they had for homework -> count the paths going from one corner to the opposite corner on a cube:

Now for a challenge -> count the paths on a 2x2x2 cube going from one corner to the other (in this case each step will have length 1):

This is a fun introductory counting exercise. I was a little surprised how difficult it was to keep track of the numbers on the final 2x2x2 cube, but it was nice to see that they boys could see how to count those paths directly with choosing numbers.

Building Paula Beardell Krieg’s cube

Yesterday we studied how to build the pieces of Paula Beardell Krieg’s dissected cube:

Paula

3d printing Paula Beardell Krieg’s dissected cube shapes

Today the shapes were done printing and I had the kids talk about them one more time:

After that short conversation I had each kid tweak the code that we used to make the shapes to make a new shape. Here’s what my younger son made:

Here’s what my older son made:

Definitely a fun couple of days with these shapes. Will probably revisit them again in a few months.

Revisiting the volume of a sphere with 3d printing

[Note: 10:30 am on Oct 7th, 2017 – had a hard stop time to get this out the door, so it is published without editing. Will (or might!) edit a bit later]

About two years I found an amazing design by Steve Portz on Thingiverse:

“Archimedes Proof” by Steve Portz on Thingiverse

Screen Shot 2017-10-07 at 9.21.36 AM

We did a really fun project using the print back then:

The volume of a sphere via Archimedes

Today we revisited the idea. We began by talking generally about the volume of a cylinder:

The next part of the project was heading down the path to finding the volume of a cone. I thought the right idea would be to talk first about the volume of a pyramid, so I introduced pyramid volume idea through snap cubes.

Also, I knew something was going a little sideways with this one when we were talking this morning, but seeing the video now I see where it was off. The main idea here is the factor of 3 in the division. Ignore the height h that I’m talking about.

Next we looked at some pyramid shapes that we’ve played with in the past. The idea here was to show how three (or 6) pyramids can make a cube. This part was went much better than the prior one 🙂

The ideas here led us to guess at the volume formula for a cone.

Now that we’d talked about the volume formulas for a cone and a cylinder, we could use the 3d print to guess at the volume formala for the sphere.

With all of that prep work behind us, we took a shot at pouring water through the print. It worked nearly perfectly 🙂

I am really happy that Steve Portz designed this amazing 3d print. It makes exploring some elementary ideas in 3d geometry really fun!

Extending our Alexander Bogomolny / Nassim Taleb project from 3 to 4 dimensions

Last week I saw really neat tweet from Alexander Bogomolny:

The discussion about that problem on Twitter led to a really fun project with the boys:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

That project reminded the boys about a project we did at the beginning of the summer that was inspired by this Kelsey Houston-Edwards video:

Here’s that project:

One more look at the Hypercube

For today’s project I wanted to have the boys focus on the approach that Nassim Taleb used to study the problem posed by Alexander Bogonolny. That approach was to chop the shape into slices to get some insight into the overall shape. Here’s Taleb’s tweet:

Taleb1

So, for today’s project we followed Taleb’s approach to study a 4d space similar to the space in the Bogomolny tweet above. The space is the region in 4d space bounded by:

|x| + |y| + |z| \leq 1,

|x| + |y| + |w| \leq 1,

|x| + |w| + |z| \leq 1, and

|w| + |y| + |z| \leq 1,

To start the project we reviewed the shapes from the project inspired by Kelsey Houston-Edwards’s hypercube video. After that we talked about the equations we’d looked at in the project inspired by Alexander Bogomolny’s tweet and the shape we encountered there:

Next we talked a bit about the equations that we’d be studying today and I asked the boys to take a guess at some of the shapes we’d be seeing. We also talked a little bit about absolute value which briefly caused a tiny bit of confusion.

The next part of the project used the computer. First we reviewed Nassim Taleb’s approach to studying the problem posed by Alexander Bogomolny. I think it is really useful for kids to see examples of how people use mathematical ideas to solve problems.

The 2d slicing was a fascinating way to approach the original 3d problem. We’ll use the same idea (though in 3d) to gain some insight on the 4d shape.

One fun thing about this part of the project is that we encountered a few shapes that we’ve never seen before!

Finally, I revealed 3d printed copies of the shapes for the boys to explore. They immediately noticed some similarities with the hypercube project. It was also really interesting to hear them talk about the differences.

At the end, the boys think that the 4d shape we encountered in this project will be the 4d version of the rhombic dodecahedron. We’ve studied that shape before in this project inspired by a Matt Parker video:

Using Matt Parker’s Platonic Solid video with kids

I don’t know if we are looking at a 4d rhombic dodecahedron or not, but I’m glad that the kids think we are 🙂

It amazes me how much much fun math is shared on line these days. I’m happy to have the opportunity to share all of these ideas with my kids!

Calculating the volume of our rhombic dodecahedron

Yesterday we did a fun project involving a rhombic dodecahedron:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

At the end of that project we were looking carefully at how you would find the volume of a rhombic dodecahedron in general. Today I wanted to move from the general case to the specific and see if we could calculate the volume of our shapes. This tasked proved to be much more difficult for the boys than I imagined it would be. Definitely a learning experience for me.

Here’s how we got going. Even at the end of the 5 min here the boys are struggling to see how to get started.

So, after the struggle in the first video, we tried to back up and ask a more general question -> how do we find the volume of a cube?

Now we grabbed a ruler and measured the side length of the cube. This task also had a few tricky parts -> do we include the zome balls, for example. But now we were making progress!

Finally we turned to finding the volume of one our our 3d printed rhombic dodecahedrons. We did some measuring and found how many of these shapes it would take to fill our zome shape and how many it would take to fill a 1 meter cube.

So, a harder project than I expected, but still fun. We’ve done so much abstract work over the years and that makes the concrete work a little more difficult (or unusual), I suppose. I’m happy for this struggle, though, since it showed me that we need to do a few more projects like this one.

Some beautiful geometry in a challenge problem from Alexander Bogonolny

I did the project below with the boys on Sunday before they went off to camp for a week. The idea wasn’t to get into heavy math, but rather just a relaxed walk through some fun shapes. We got one detail wrong in the 4th video which I was sort of kicking myself for, but then I saw a tweet from Nassim Taleb showing some of the geometry in a different problem that Alexander Bogonolny had posted and it made me realize the connection between the algebra and geometry in our problem was still fun to show:

So, despite the error I thought I would publish the project anyway.

Here’s the original problem:

Below are the videos showing our walk through the geometry. First, though, here’s the quick introduction to the problem:

After that intro we looked at the region described by the constraint in the problem. We have to thicken up the region a little bit using the absolute value function in order to see it, so the Mathematica code looks a bit more complicated than in the problem, but that extra complexity is just to make the picture easier to see.

One cool thing about our discussion here is that my younger son thought there should be 3 fold symmetry in the shape because there was 3 fold symmetry in the equation 🙂

Now we looked at the situation in which the surface achieves the maximum value subject to the constraint in the problem. My younger son made the nice observation that the two surfaces appeared to be “blending together” at certain points. That “blending” is an important idea in Lagrange Multipliers – though, don’t worry, we aren’t going down that path today.

Next we looked at the minimum value of the surface subject to the constraint in the problem. The error I made here was accidentally reversing the two surfaces. The fixed surface – the one describing the constraint – is now on the outside rather than the inside.

Finally, I asked the kids to pick a value smaller than 45/4 for the curve so that we could see what happened. Unfortunately they picked 7 which is too small – there’s no surface! – so they chose 10 and that allowed us to see that the shrinking surface inside of the original shape. Also we can see fairly clearly (after some rotation) that the two shapes do not intersect.

Definitely a fun project showing the boys a beautiful side of a really challenging problem.

Working with the PCMI books part 2: coloring an octahedron

Last week we got the PCMI books:

Our first project involved a neat problem about understanding the number 0.002002… in different bases:

Playing around with the PCMI books

Today I was looking for another fun problem and found another problem that I thought would make a fun project:

Barbara has an octahedron, and she wants to color its vertices with two different colors. How many different colorings are possible? By “different” we mean that you can’t make one look like the other throu a re-orientation.

I started by introducing the problem and asking the kids what their initial ideas were:

They had a couple of pretty good ideas including some basic ideas about symmetry. Using those ideas we began counting the different colorings:

We counted the cases in which 3 vertices were black and 3 vertices were red. This case proved to be tricky, but going through it slowly got us to the correct answer.

Finally, as a fun little extension, I asked them to find the number of ways to color the faces of a cube with two colors. Having solved the octahedron problem already, this one went pretty quickly, and they even noticed the connection between the two problems 🙂

I like this problem. I’m glad that the boys were able to see some of the basic ideas. When you add more colors the counting gets much more difficult and some pretty advanced math comes into play. The number of colorings with “n” colors is:

(n^6 + 3n^4 + 12n^3 + 8n^2) / 24

The different terms correspond to different symmetries of the cube / octahedron. We’ll have to wait a few more years to cover the complete details 🙂

Working through “Euler’s Gem” with kids

A few weeks ago I stumbled on Diave Richeson’s book Euler’s Gem:

Although the book is not intended for kids, it is written for a general audience and I thought the boys might enjoy working through the book slowly. I’ve been having them read one chapter per day and they are really having fun with the ideas.

Today I asked each of them to talk about what they’ve learned through the first 5 chapters. Here’s what my younger son had to say:

Here’s what my older son had to say:

It is fun to hear what they are taking away from this book and also really nice to hear that both of them really do like the book. I’ve not tried an experiment like this one before, but the book is so well written that I really do think that with a little bit of help here and there kids can understand most of it.