# 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. Β 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

# Exploring Annie Perkins’s Cairo Pentagons with kids

I saw a great tweet from Annie Perkins a few days ago:

I thought it would be a fun idea to add to the list of our growing list of pentagon projects. At this point I’ve lost track of all of them, but they got started with this amazing tweet from Laura Taalman:

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

and the most recent project (I think!) is this one:

Evelyn Lamb’s pentagons are everything!

Oh, and obviously don’t forget pentagon cookies π

After seeing Perkins’s tweet I started down the path of making the Cairo tiling pentagons but super unluckily had a typo in my printing code. At least my cat made good use of the not-quite-Cairo pentagons:

So, while I wait for the correct pentagons to print, I thought I’d talk about the special shape of the Cairo tiles with my older son. One of the neat things about all of these pentagon projects is getting to talk about geometry with kids in sort of non-standard, non-textbook way. Tonight’s conversation was about coordinate geometry using the properties of the Cairo pentagon.

Wikipedia’s page on the Cairo tiling pentagon

To start the project I drew the shape on our board and asked my son to find the coordinates of the points. This is a bit of an open ended question because you have to know the lengths of the side so know the coordinates – I was happy that he noticed that problem (and, just to be 100% clear, I don’t know for sure if there are restrictions on the sides for the Cairo tiling – I’ll learn that when the new pentagons finish printing – ha ha).

Here’s how he started in on the problem:

For the second part of the project he had to make one more choice for a side length, and then he was able to find the coordinates of all of the corners of the pentagon.

One of the great (and happy) surprises with math and 3d printing is that you get neat opportunities to explore 2d geometry. Some of our old projects exploring 2d geometry with 3d printing are here:

Using 3d printing to help kids learn algebra and 2d geometry

I’m excited to play with the Cairo tiles when they finish printing tonight. Hopefully the 2nd time is a charm!

# Thinking about a math appreciation class

Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:

I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:

Here’s a link to that set of projects:

Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)

So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic.Β  I chose folding and thought about what sort of ideas could be shared with students.

Below are 9 ideas that came to mind along with 30 second videos showing the idea.

(1) A surprise book making idea shown to me by the mother of a friend of my older son:

(2) Exploring plane geometry through folding

We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:

Our Patty Paper geometry projects

Here’s one introductory example showing how to find the incenter of a triangle:

(3) The Fold and Cut theorem

Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:

OUr Fold and Cut projects

Here’s one fun fold and cut example:

(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets

You can find Taalman’s post about these hinged polyhedra here:

Laura Taalman’s hinged polyhedra blog post on her Makerhome blog

And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!

Which comes from this amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

[space filled in with random words to get the formatting in the blog post right π ]

(5) An amazing cube dissection made by Paula Beardell Krieg

We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:

Our projects based on Paula Beardell Krieg’s work

(6) And Paula didn’t just stop with one cube π

(7) Two more of Laura Taalman’s prints

Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.

Here’s Taalman’s blog post about the Peano curve:

Laura Taalman’s peano curve 3d print

(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke

As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:

(9) Current research by Laura DeMarco and Kathryn Lindsey

Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.

Quanta Magazine’s article on DeMarco and Lindsey’s work

The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids

So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.

# Evelyn Lamb’s pentagons are everything

Last week Evelyn Lamb published a fantastic article:

Math Under My Feet

In a way – a super serious way – I don’t want you to read this blog post. I want you read her article and just think about some of the properties that the tiling pentagons in article probably have.

The question that same to my mind was this one -> Why are the pentagons in her article Type 1 pentagons?

The resources I used initially to help with this question were:

(i) the pictures of the different tiling patterns in the article:

(ii) Laura Taalman’s Tiling Pentagon resource on Thingiverse:

(iii) and then when I was stumped and wrote to Evelyn she pointed me to the Wikipedia page for tiling pentagons – which is really good!

Wikipedia’s page on pentagon tilings

So, honestly, stop here and play around. You don’t have to have the nearly week long adventure with these pentagons that I did, but I promise that you will enjoy trying to figure out the amazing properties of this damn shape!

If that adventure is interesting to you, I think you’ll also find that sharing that adventure with students learning algebra and geometry would be pretty fun, too!

Here are some of our previous pentagon tiling projects:

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

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

Also, here are the first two projects that I did with the boys after reading Evelyn Lamb’s latest article:

Evelyn Lamb’s Tiling Pentagons

Using Evelyn Lamb’s tiling pentagons to talk about lines and shapes with kids

The problem with those last two projects is that they aren’t actually right. I hadn’t properly understood the shape . . . dang π¦

With a bit more study, though, I did *finally* understand this damn shape!!

So, I printed 16 of them and set off on one more project with the boys tonight. The goal was to show them the 3 completely different tilings of the plane that you can make with Evelyn Lamb’s pentagon.

I won’t say much about the videos except that watching them I hope that you will see that (i) this is a great way to talk about geometry with kids (building the shapes is a great way to talk about algebra), and (ii) that understanding these tiling patterns is much harder than you think it is going to be. As an example of the 2nd point, it takes the boys nearly 10 minutes to make the tiling pattern in Lamb’s article.

So, here’s how things went:

(1) An introduction to the problem:

(2) Using the pentagons to make the “standard” Type I tiling pattern

This tiling pattern is in the upper left hand corner of the picture above that shows the collection of pentagon tiling patterns.

Β

(3) Using the pentagons to make the “pgg (22x)” tiling pattern from the Wikipedia article:

(4) Part I of trying to make the tiling pattern in Evelyn Lamb’s article:

(5) Part 2 of Evelyn Lamb’s tiling pattern:

Don’t really know what else to add. I think playing around with the math required to make these pentagons AND playing with the pentagons themselves is one of the most exciting algebra / geometry projects for kids that I’ve ever come across.

I’m so grateful for Evelyn Lamb’s article. It is really cool to see how a mathematician views the world and it is so fun to take her thoughts and ideas and turn them into projects for kids

# 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 π

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.

# Playing with some 3d printed knots

Today we looked at some 3d printed knots designed by Laura Taalman and Henry Segerman.

Two are versions of Taalman’s “rocking knot” which we found here:

Laura Taalman’s Makerhome blog: Day 110 – the Rocking Knot

The second is the Torus knot from Segerman’s new book Visualizing Mathematics with 3D Printing.

We started the project today by just talking about the knots. Comparing the two knots that are actually identical was useful in refining the language they used to talk about knots.

Next they wanted to try to compare the two identical knots by looking at their crossings. My older son had the idea of assigning a +1 to every “over” crossing and a -1 to every “under” crossing. My younger son noticed that this counting method should always produce a net 0 because we counted the over and under crossing for each crossing exactly once.

New we tried to compare Segerman’s torus knot to Taalman’s rolling knot. Here we used the “tangle” from Colin Adams’s book Why Knot?

One fun thing that came up by accident in this video is an amazing shadow cast by Taalman’s knot – that was a really fun surprise.

Unfortunately, it proved to be a bit difficult to get the tangle back together so we had to pause the video at the re-connect the tangle off camera. It is really neat, though, to watch kids try to make a copy of a knot.

Once we got the tangle connected we started the next video. Since the tangle can move around, it isn’t that hard to manipulate the tangle from the form Segerman’s knot to the form of Taalman’s knots. In fact, it happened more or less by accident!

As I mentioned above, it is actually a pretty difficult task for the kids to describe the features of the knots when they compare them – even with a knot as simple as the trefoil knot. I think one of the neat parts of this particular project is working on using more precise mathematical language.

So, a fun project. We have a new 3d printer and I’m really excited about using many more 3d printing ideas from Taalman and Segerman to explore math with the boys.

# Playing with Laura Taalman’s Peano Curves

I’m starting to think about what to do for the Family Math nights at my younger son’s school this year. During the day today I 3d printed 2 of Laura Taalman’s Peano Curves to see if that might somehow make a fun project for a group of 4th and 5th graders.

Taalman’s blog post about the curve is here:

Laura Taalman’s 3d-printed Peano curve blog post

The plan tonight was to have each kid talk about the curve (they’ve seen it before) and see what they thought was interesting. My older son went first:

Then my younger son:

For the last part of the project we took the curve off the base and stretched it out (almost) into a line:

I think there’s a fun project here – these take a long time to make, but I think with a week or two of printing prep that there’s a good 45 minute project for 4th and 5th graders in here somewhere.

# Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

You can find Taalman’s program here:

and our project is here:

During the project my younger son found a different tiling pattern for pentagon #10 than the one in Taalman’s program. I suspected that the tiling pattern actually related to Pentagon #1 but wasn’t sure.

When the 15th tiling pentagon was discovered last year Evelyn Lamb wrote this great article which mentioned that each pentagon was actually part of an infinite family of tiling pentagons:

Tonight I used Taalman’s program to show my younger son how to make his tiling pattern from pentagon #1.

First I had him recreate the two tiling patterns:

Next we used the amazing functionality in Taalman’s tiling pentagon program to find this tiling pattern in pentagon #1:

So, thanks to Laura Taalman and Evelyn Lamb for teaching us something about tiling pentagons tonight!

Last year a 15th tiling pentagon pattern was discovered ( See this incredible article by Evelyn Lamb for more info) and Laura Taalman showed how to 3d print all of the patterns:

Her print patterns went well beyond just plain old pentagons, though.Β  She even included cookie cutter versions that we used for a really fun project for kids:

Using Laura Taalman’s 3D printed pentagons to talk math with kids

You know what we never did with those cookie cutters, though . . . ACTUALLY MAKE COOKIES.

I was reminded of that terrible failure when I saw this really cool video about shapes from Eugenia Cheng last week:

After watching the video I wrote up a quick post about how you could extend a few of the ideas that Cheng discusses.

Extending Eugenia Cheng’s “shapes” video

Today it was time to make cookies!

We started by watching Cheng’s video (the kids were on vacation with their cousins last week, so they hadn’t see it) and reviewing Taalman’s 3D printing site on Thingiverse. Oh, sorry about the hiccups . . . :

Yesterday I had the boys each pick a pentagon to play with. Using the numbering in Taalman’s project my younger son picked #10 and my older son picked #8. I printed 24 of each pentagon and had the boys play around and try to discover the tiling pattern.

Here’s my older son discussing the tiling pattern for #8 which was actually very difficult to find:

Here’s my younger son talking about finding the tiling pattern for pentagon #10. I got a bit of a surprise when he found a tiling pattern that was completely different than the tiling pattern that Taalman showed for #10.

I think that this different pattern is actually part of the family of pentagons from pentagon #1 in Taalman’s list, but I’m not sure. It was definitely fun that he found an alternate way to tile with this pentagon.

We finished up with what was obviously the most important part of the project – making cookies! Here are the cookies being cut out. Unfortunately the tiling pattern with pentagon #8 needs a flipped over version, so we didn’t think we could make the tiling pattern with the cookie cutter we had.

The patterns for #10 both work, though, and my younger son made each of them:

So, a great project today thanks to Laura Taalman and Eugenia Cheng. Can’t wait to try out the cookies!

# 120-sided dice!!

Our d120s arrived!!

Here’s the unboxing:

We were pretty lucky to have gotten our order in before all of the publicity caused the number of orders to explode. Before the boys ran off to show their friends the new dice, we did a few projects. I asked the boys to think of a question that they thought would be interesting to study with the dice. They didn’t have a lot of time to think about it, but I just wanted their gut reactions anyway.

My older son thought it would be fun to see how far they rolled. We have several different types of dice around the house, plus a few 3d printed shapes, so we saw how far the different shapes rolled:

The “winner” wasn’t actually one of the d120s, but rather a pentagonal hexecontahedron that we’d printed from Laura Taalman’s blog:

Day 194 of Laura Taalman’s Makerhome blog – the Pentagonal Hexecontahedron

The Dice Lab actually makes a d60 in the shape of a deltoidal hexecontahedron, so – no surprise, really – they are way ahead of us!

My younger son wanted to use them to make binary codes. I didn’t quite understand what he meant, but we eventually decided that the odd numbers would represent a 1 and the even numbers would represent a 0. We rolled the dice to create some 5-digit binary numbers. Strangely, we rolled lots of even numbers:

Finally, we did a project that my wife suggested – how many rolls to you think it will take until we see a number that we’ve already rolled? Fun! We had a great time exploring this question.

So, some fun little projects with the dice. Now the boys are off showing their friends and using them for some advantage playing Magic: The Gathering π

Oh, and just in case you’ve not seen the video about these new dice, here it is: