Evelyn Lamb’s tiling pentagons

Since the 15th tiling pentagon was discovered in 2015 we’ve done some fun projects with tiling pentagons. A key component in all of our project was Laura Taalman’s incredible work that made all 15 pentagon tilings accessible to everyone:

Here are a few of those projects:

Using Laura Taalman’s 3d Printed Pentagons to talk math with kids

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

and, of course, pentagon cookies ðŸ™‚

Evelyn Lamb has also written some absolutely fantastic articles on tiling pentagons. Here original article on the subject was critical in helping me understand what was going on in the different tilings:

There’s Something about Pentagons by Evelyn Lamb

And her amazing article from last week (April 2017) inspired today’s project:

Math Under My Feet

The prep work for this project was probably 100x more than I usually do because the tiling described in Lamb’s article turned out to be very hard for me to understand. It didn’t look like the “type I” tiling pictured in the article and I spent days trying to see if it was somehow a sneaky form of one of the other tilings.

Finally I wrote to Lamb and asked her about it and she pointed me to the Wikipedia page here which showed that the type 1 tilings have two different forms. One form has a repeating pattern with 2 pentagons and the other has a repeating pattern with 4 pentagons. Ahhhhhh – at last I saw what I was missing and why this “new to me” type 1 tiling was so elusive:

Wikipedia’s page on pentagon tilings

So, having finally understood what was going on with this octagon / pentagon tiling, I got to work making some of the pentagons. I didn’t quite match the pentagons in Lamb’s article, but the ones I made still have the property that they can produce two different tilings.

I got started this morning by having the kids read Lamb’s new article. Here’s what they thought:

Next I had the boys try to make a tiling from the pentagons I made last night. They made the first type of tiling (the one that has two repeating pentagons) and we talked about whether or not that was the tiling in Lamb’s article.

I include the whole process of finding the tiling here to show that even a tiling with two repeating pentagons isn’t so easy to find as you might think.

Now we went to the both Lamb’s article and to the Wikipedia pentagon tiling page to study the various different types of Type I tilings. I’m still a little confused as to what makes tilings different, but however the classification works, here’s our discussion of the various Type I tilings.

Off camera I had the boys try to make the new type of tiling. It took a while (though not super long – from the time they started reading the article until the time we finished the project was roughly 30 min).

Once they had the tilings I turned on the camera to talk about the shapes:

This was such a fun project! Tomorrow I hope to do a second project to show how making these pentagons is a great way to help kids learn about / review basic properties of lines.

Sharing Grant Sanderson’s Calculus Ideas video with kids

Yesterday I saw an incredible new video from Grant Sanderson:

As is the case with all of his videos, this one totally blew me away. I also thought that it has some fantastic ideas to share with kids. So, this morning we tried it out!

I started by asking the boys about the area of a circle – how do you find the area?

We have studied the idea before. Here’s the previous idea (that we got from a Steven Strogatz tweet):

and here are the projects inspired by Strogatz’s tweet:

Steven Strogatz’s circle-area exercise

Steven Strogatz’s circle-area project part 2

Fortunately, the boys were able to remember that idea and explain it pretty well:

After this short discussion I had the boys watch the new “Essence of Calculus” video. I actually left the room so that I wouldn’t interfere. The video below shows the ideas that they found interesting. One thing – luckily! – was the idea of making lots of slices and getting a better and better approximation to a shape. We were able to connect that idea to our prior way of finding the area of a circle, which was nice ðŸ™‚

Next we talked about the new (to them) way of finding the area of a circle that Sanderson explains in his video. What made me really happy here is that my younger son was able to understand and explain most of the ideas. It think that a 5th grader being able to grasp these ideas really shows the tremendous quality of Sanderson’s explanation in his video. I also think that it shows that many important ideas from advanced math are both accessible and interesting(!) to kids

Finally, I showed the boys some 3d prints that I made overnight.

These prints were pretty easy to make and I hoped that they would make some of the approximation ideas seem more real. In the middle of the video I remembered that I’d actually tried this idea before (ha!):

3d printing and calculus concepts for kids

After remembering the old project, I ran and got the old shapes, too:

I’m really excited for the rest of this new calculus series. Some of the more advanced ideas might not be so great for kids, but I hope to share one or two more with the boys just to show them a few ideas that they’ve probably never seen before. Plus – I’ve got no doubt at all that this whole series is going to be amazing!

Comparing Sqrt(x^2 + y^2) and ( Sqrt(x^2) + Sqrt(y^2) )

Last week we used 3d printing to compare $(x + y)^2$ and $x^2 + y^2$:

That project is here:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing

My younger son is still sick today and not able to participate in a math project, so I chose a slightly more algebraically complicated comparison to look at with just my older son -> $\sqrt{x^2 + y^2}$ and $\sqrt{x^2} + \sqrt{y^2}$

Here’s what the shapes look like:

I started the project by reviewing the original project in this series just to remind my son about how we thought about the 3d surfaces in the prior post. He remembered most of the ideas, fortunately, so the introduction was fairly quick.

After the introduction we talked about some basics of the algebra we were going to encounter in this project, namely that $\sqrt{x^2} = |x|$. This part all by itself is a difficult concept to understand and the bulk of the video below was spent talking about it.

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With the difficult part of the algebra behind us we moved on to talking about the surface $z = x^2 + y^2$. What does this surface look like?

I really enjoyed the discussion here – the question is actually a pretty challenging one for a kid to think through.

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Next we tried to figure out what the surface $z = \sqrt{x^2} + \sqrt{y^2}$ would look like.

I think it takes a while to get used to working with graphs of the square root function. My son struggled a bit here to figure out the shape here. Hopefully that struggle helped him

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Now I revealed the shapes and let my son discuss the properties of the shapes now that he could hold them in his hand. There were a few surprises, which was nice ðŸ™‚

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I’m really happy about this series of projects. It is fun to explore the variety of ways that 3d printing can help kids explore math.

Comparing x^2 + y^2 and (x + y)^2 with 3d printing

Yesterday we did a project exploring a common algebra mistake -> assuming that $(x + y)^2 = x^2 + y^2$. That project is here:

Does (x + y)^2 = x^2 + y^2

Today I thought it would be fun to explore the same idea using 3d printing. During the day I made prints of the two surfaces

(i) $z = x^2 + y^2$, and

(ii) $z = (x + y)^2$.

Here they are:

For the project I asked the boys to try to figure out what the two graphs looked like over the domain -2 < x < 2, and -2 < y < 2, and they showed them the shapes. Though not really by design, the choice of a square for the domain turned out to lead to an interesting discussion at the end.

Here's how the project went:

(1) What does $z = x^2 + y^2$ look like?

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(2) What does $z = (x + y)^2$ look like?

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(3) What about the actual shapes surprises you?

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I really enjoyed the combination of these two projects. Hopefully seeing the shapes of the surfaces becomes one little extra reminder that the two commonly confused expressions $x^2 + y^2$ and $(x + y)^2$ are not the same.

Learning 3d geometry with Paula Beardell Krieg’s pyramids

Earlier in the week we got a nice surprise when we received a fun little pyramid puzzle from Paula Beardell Krieg:

Our initial project using the shapes is here:

Playing with an amazing present from Paula Beardell Krieg

I thought a follow up project would be fun, so I decided to try out a basic exploration in 3d geometry. The goal was to make these shapes ourselves using Mathematica, then to 3d print them, and finally to play with the new shapes to see that they were indeed the same.

We started by talking about the shapes in general and see if we could identify some very specific properties of the shape using coordinate geometry:

Next we talked about how to describe the planes that formed the boundary of the shape. It was fun hearing my 5th grader try to figure out how to describe the planes (and regions) we were studying here. One other challenge here is that we were also trying to describe the 3d regions above and below these planes.

Now came the special challenge of finding a mathematical way to describe the hard to describe plane in the shape. I had to guide the discussion a bit more than I usually do here, but the topic of finding the equation for a plane is pretty advanced and something that kids have not seen before.

Having written down the equations, we went up to look at the Mathematica code I’d used to make the shapes. The boys were able to see that the first shape had exactly the same equations we’d written down, and they were able to see that the equations for the 2nd shape were not any more difficult.

The shapes printed overnight and we had an opportunity to play with them this morning. It is pretty neat to hear them compare the shapes and see that, indeed, the shapes we made are really the same as the shapes Paula sent us.

So, there’s quite a lot we can study with Paula’s shapes. You’ve got the potential to study folding patters, basic 3d geometry, the volume formula for a pyramid, and even 3d printing! Fun how such a seeming simple idea can lead you in so many different directions.

Playing with shapes of constant width

Tonight’s project was just for fun – they boys both had long days of school / music.

So, last night I downloaded and printed some shapes of constant width from Thingiverse:

Anenome’s Object of Constant Width on Thingiverse

With 4 of them on the table I asked each of the boys what they thought the shapes were and then let them play around with them. After they played for a bit I put a book on the shapes and asked them how they thought the book would move as the shapes rolled.

Here’s what my younger son thought:

Here’s what my older son thought:

I always find it fun to hear what kids think about complicated shapes. Lots of neat ideas and then a good “wow” when you learn the secret property!

A fun shape for kids to explore: the Permutohedron

I learned about permutohedrons from a comment by Allen Knutson on a prior blog post. See the first comment here:

A morning with the icosidodecahedron thanks to F3

I prnted the shape from Thingiverse and it was amazing!

“Permutahedron” by PFF000 on Thingiverse

We started the project today by examining the shape and comparing it to a few other shapes we printed. The comparison wasn’t planned – the other shapes just happened to still be on the table from prior projects . . . only at our house ðŸ™‚

Next we talked about permutations and the basic idea we were going to use to make the permutohedrons. We drew the 1 dimensional version on the whiteboard and talked about what we thought the 2 dimensional version would look like.

We used our zometool set to make a grid to make the 2 dimensional permutohedron. Lots of different mathematical ideas for kids in this part of the project -> coordinate geometry, permutations, and regular old 2d geometry!

Next we went back to talk about how PFF000’s shape was made. Here’s the description on Thingiverse in case I messed up the description in the video:

“The boundary and internal edges of a 3D permutahedron.

The 4! vertices are given by the permutations of [1, 3, 4.2, 7], with an edge connecting two vertices if they agree in two of the four coordinates. The 4D vertices live in a 3D hyperplane, namely the sum of the coordinates is 15.2.

This part of the project was a little longer, but worth the time as both the simple counting ideas on the shape and the combinatorial ideas in the connection rules are important ideas:

Finally we wrapped up by taking a 2nd look at the shape and also comparing it to Bathsheba Grossman’s “Hypercube B” which was also still laying around on our project table!

This was a really fun project that brought in many ideas from different areas of math. I’m grateful to Allen Knutson for the tip on this one!

Sharing a shape from calculus with kids

Finding the volume of the intersection of two cylinders is a common calculus problem. The shape also plays a role in this old (for the internet!) video from Brooklyn tech that inspired me to get a 3d printer:

Today for a fun project to start the week I decided to share the shape with the boys and see what they thought about it. My younger son went first:

After playing on the computer I had him explore the printed version of the shape – make sure to stay to about 1:25 to hear where he thinks this shape might occur in “real life” ðŸ™‚

Next my older son played with the shape on the computer. He remembered seeing it before in a project from a month ago on the intersection of 3 cylinders:

Exploring 3 intersecting cylinders with 3d printing

Next he played with printed shape. I asked him to describe how he thought you’d be able to figure out that the shape was made out of squares – I thought his answer was pretty interesting. This question gets to the math ideas behind the calculus problem.

It is sort of fun for kids to see and play with shapes like this – no need to wait for calculus anymore to explore interesting shapes!

A morning with the icosidodecahedron thanks to F3

A few weeks a go I saw this shape in a display case at the MIT math department:

The shape is mislabled, unfortunately, it is an icosidodecahedron. We’ve already done a few projects based on the shape. Last week’s project is here:

A zometool follow up to our Cuboctohedron Project

And there’s our fun Zometool Snowman, too, where the icosidodecahedron is the head:

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The name suggests that it is made from a combination of an icosahedron and a dodecahedron – but how?

(i) the icosidodecahedron
(ii) a dodecahedron with an icosahedron removed
(iii) an icosahedron with a dodecahedron removed

Here’s what the boys thought of those shapes:

Next we went upstairs to play with some code in the F3 program. Looking at the video now I see that I forgot to publish it hi def – sorry about that. I hope our explanation of the code is good enough if the code is too fuzzy to read:

Definitely a fun little project – it is so fun to be able to play with these shapes on the computer and then hold them in your hand!

Using 3d printing to review properties of lines

As a follow up to our last two projects with 3d printed triangles, I thought it would be fun to try out a similar project. The point of the 2nd project wasn’t the geometry, though, it was to use the process of making the shapes as a way to review some basic ideas about lines.

So, I started by showing my older son the basic idea – we wanted to write down the equations of the three lines that border a 5-12-13 triangle. Since we have a right triangle the equations aren’t too difficult, but are still useful for a simple review:

Next we went upstairs to Mathematica to make our 3d template using the function RegionPlot3d:

After the prints finished we played around with them a bit to see which triangles we could make with the same area:

I think that making little shapes like this might be one of the best educational uses of 3d printing. Kids get an opportunity to apply some basic math knowledge and create some fun shapes!