Sharing a Craig Kaplan post with kids

I saw the latest post from Craig Kaplan via a tweet from Patrick Honner:

The picture in the middle part of the post looked like something that kids could understand:

For our project today I thought it would be fun to talk about how to make the polygon tile in the above picture. After we understand how to describe that polygon, we can 3d print a bunch of the tiles and talk more about the idea of “surrounding a polygon” with these tiles tomorrow.

This project is a fun introduction to 2d geometry (and especially coordinate geometry) for kids. We also use the slope / intercept form of a line when we make the shape.

We got started by looking at Kaplan’s post:

Next we began to talk about how to make the shape – the main idea here involves basic properties of 30-60-90 triangles. My older son was familiar with those ideas but they were new to my younger son.

We also talk a little bit about coordinate geometry. The boys spend a lot of time discussing which point they should select to be the origin.

In the last video we found the coordinates of 3 of the points. Now we began the search for the coordinates of the other two. We mainly use the ideas of 30-60-90 triangles to find the coordinates of the first point.

The 2nd point was a bit challenging, though:

The next part of the project was spent searching for the coordinates of the last point. The main idea here was from coordinate geometry -> finding the coordinates of the middle of the square. The coordinate geometry concepts here were difficult for my younger son but we eventually were able to write down the coordinates of the final point:

We were running a little long in the last video, so I broke the video into two pieces. The last step of the calculation is here:

After finding all of the coordinates we went upstairs to make the shape on Mathematica. We used the function “RegionPlot3D” that allows us to define a region bordered by a bunch of lines. Below is a recap of the process we went through to make the shape and a quick look at the shapes in the 3d printing software:

This isn’t our first 3d printing / tiling project. Some prior ones are linked in a project we did last month after seeing an incredible article by Evelyn Lamb:

Evelyn Lamb’s pentagons are everything

I’m excited with the boys to play with the tiles from Kaplan’s post tomorrow.

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 😦

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.

(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 🙂

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.

A zipper Möbius strip from Mathjams

Saw this tweet yesterday:

A few weeks ago we’d done another Möbius strip-related project I saw in one of James Tanton’s books:

An absolutely mind-blowing project from James Tanton

After seeing the re-tweet from Mathjams I thought it would be fun to try this project, too. I ran to Joann Fabrics and found some 48 inch zippers and my wife helped me figure out how to make the shape. She did the sewing and commented that she was surprised that the pins we had holding the shape together started on once side and ended up on the other side! So, there’s a neat “what do you notice” math and sewing project here, too. See also:

How to sew like a mathematician by Evelyn Lamb

I had each kid go talk about the shape separately so that the other kid’s ideas wouldn’t spoil theirs. Here’s what my younger son had to say as he played with the shape:

and here’s what my older son had to say:

I love this shape and exploring it is a great math project for kids (and probably for everyone!). Thanks to Mathsjam and to Andrew Taylor for sharing it.

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.  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!

Sharing math with the public and especially with kids

My wife and kids are up hiking in New Hampshire this weekend and I’m home with a cat who misses the kids. Yesterday I was watching Ed Frenkel’s old Numberphile interview about why people hate math:

The line about 50 seconds in to the video has always really resonated with me – “How do we make people realize that mathematics is this incredible archipelago of knowledge?” As has the his point later in the video that mathematicians have not generally done a great job sharing math with the public (say from 5:00 to 6:30).

Frenkel’s piece has played a role in many of my blog posts, here are three:

Sam Shah – a high school teacher in New York – wrote a great piece about sharing math that is not typically part of a high school curriculum with kids, and gave some suggestions for projects:

A Partial Response to Sam Shah

Lior Pachter wrote an incredible blog post about sharing unsolved math problems under the Common Core framework. I copied his idea but used math from mathematicans rather than unsolved problems:

Sharing math from Mathematicians with the Common Core

Then when the sphere packing problem was cracked by Maryna Viazovska earlier this year, I wrote about how this was a great opportunity for mathematicians to share a math problem with the public:

A challenge for professional mathematicians

As you can tell, I watch Frenkel’s video quite a bit 🙂 While I was watching the video yesterday I received this message:

Though he isn’t a professional mathematician, this article from Brian Hayes is really close to what I’d love to see from mathematicians.

As are the articles by writers like Erica Klarreich and Natalie Wolchover at Quanta Magazine:

Quanta Magazine’s math articles

and mathematician Evelyn Lamb who has somehow found the time to write more than 150 articles on her “Roots of Unity” blog for Scientific American:

Evelyn Lamb’s blog on Scientific American’s website

There were probably at least 10 to pick from, but here’s an example of how I’ve used one of Lamb’s pieces to talk a little bit about topology with my kids:

Using Evelyn Lamb’s “Infinite earring” article with kids

So, with that all as introduction (!) I was very excited to see Steven Strogatz share an article from Rich Schwartz last night:

I really enjoyed our project with Schwartz’s “Really Big Numbers” and I’m happy to see that he’s writing more about sharing math with kids. Hopefully Schwartz’s article will inspire a few more mathematicians to share some fun math with kids (or with the public in general). I’d love to expand this list of projects beyond 10 🙂

A fun prime problem I saw in an Evelyn Lamb tweet

Saw this sequences of tweets because of Evelyn Lamb yesterday:

It was a fun problem to play around with and, frankly, solving it was the most impressive thing I did at the gym today so I’ve got that going for me . . . .

My younger son was home sick today, but he was feeling better this afternoon so I thought it would be interested to see what he thought about the problem. It turned into a really nice talk about numbers and arithmetic.

Here’s the introduction to the problem and his first 5 minutes of work:

and here’s his work up to the solution:

This is a really great problem to get kids talking about numbers, arithmetic, and primes. Testing whether the various multiples of 6 are near a prime is a good challenge for kids, too!

Using Matt Parker’s Menger Sponge video to talk fractions with kids

Saw Matt Parker’s latest video via an Evelyn Lamb tweet yesterday:

Here’s the video:

Watching the video last night I thought that there’d be a lots of different ways to use this one with kids. I chose to use it as a way to talk about fractions and scaling. Kids, I think, will be surprised by the result involving $\latex \pi$ but diving into that part is probably too much for kids. They can appreciate the result, though, and the discussion of fractions and scaling leads right up to the Wallis formula.

I started by asking the kids what they thought about Parker’s video. We talked about their thoughts as well as a few other fractal shapes that the knew:

Next we talked about Sierpinski’s carpet and walked through the calculations for the area. The boys saw the area essentially as a subtraction problem, so I spent a lot of time trying to help them understand how to see it as a multiplication problem:

I think that the boys still didn’t really see the area change from one step to the next as a product, so we spent a little more time talking about that idea in the modified Sierpinski’s carpet that Parker talks about in the video. The way the area changes from step to step is a great fraction problem for kids.

I’m sorry that we got a little bogged down in the calculations here, but I really wanted to be sure that the kids saw the relationship between the subtraction and multiplication approaches to calculating the area.

Finally, we looked at the complete calculation for the area of this modified Sierpinski carpet. The boys noticed a pattern in the products, which was cool, and we were able to transform our product into the famous Wallis product.

So, a fun project for kids. Parker’s video is great and serves as a great motivation for diving into the calculations. The calculations themselves are a great exercise in fractions for kids.

I’d like to try to work through the 3d version, too, so maybe we’ll do that tomorrow. My back of the envelope calculation tells me that the 4d version doesn’t have the same property of having the “volume” of the 4-d sphere (which is $\pi^2 / 2$, though I may not have done the calculations correctly the first time around and will revisit them later this weekend, too.

My fun interaction with prime numbers this week

Last week I saw a amazing new result about primes by two mathematicians at Stanford – Robert J. Lemke Oliver and Kanna Soundararajan – via an Evelyn Lamb article:

Peculiar Pattern found in “Random” Prime Numbers by Evelyn Lamb

Erica Klarreich at Quanta magazine also wrote a fantastic article about the result:

Mathematicians Discover Prime Conspiracy by Erica Klarreich

and there’s also a neat discussion of the result on Terry Tao’s blog:

Terry Tao’s blog post about the new result

After seeing the two articles (I only saw Tao’s blog post today) I thought it would be fun to play around with some similar ideas and chose to look at the last digits of triples of consecutive primes. Over the course of the week I was able to use a simple program in Mathematica to count how often the different triples of last digits occur in consecutive primes in the first 10 billion primes. Right from the start I found something I didn’t expect – counting the occurrence of the triples of last digits seemed to pair the sets of last digits quite naturally into groups of 2.

For example, for 3 consecutive primes in the first 10 billion prime numbers the last digits (3,7,1) occur 178,500,881 times and the last digits (9,3,7) occur 178,500,928 times. Another example of the strange grouping is that the triple (1,1,3) occurs 147,750,170 times and the triple (7,9,9) occurs 147,761,746 times. Weird – what’s causing this clustering?

All of my data is in the google doc linked below. I’m sorry that the data in the google doc isn’t organized very well – I was just playing around for myself, but thought that it might be fun to share anyway:

My google doc with all of the data I collected this week

I didn’t really study any number theory in college or graduate school, so I have essentially no way to know if something like the counts for the last digits of consecutive prime triples pairing up is an easy to prove fact or an impossible to prove fact. After thinking about the strange groups of two for a few days without having any decent ideas I sent an e-mail to authors of the new paper and asked them for help. They wrote back last night – which was super cool! – and provided a (possibly) easy way to think about it. I sort of can’t believe that they wrote back, but I’m really excited to spend a bit more time trying to understand their explanation.

Receiving their e-mail got me even more interested in / excited about their paper, so I spent several hours today going through it one more time. The results and conjectures are general enough to apply to the problem of consecutive triples and that led me to try to see if the paper could help me get a better understanding of the data I’d collected. Happily, I was able to understand a bit more of the paper the 2nd time through,

With sort of an “I know enough to be dangerous” understanding I attempted to predict the number of various prime triples in the next set of 1 billion primes (so, last digits of three consecutive primes from the 10 billionth prime number to the 11 billionth prime number). My guesses are in column R and column U of the “Approximations” tab in my google doc. The results should be in tomorrow morning 🙂

One fun thing about the two sets of guesses is that the sum of the guesses for all of the triples adds up to almost exactly 1 billion! Since I’m looking at 1 billion primes the sum be 1 billion, but I didn’t take that constraint into account (not directly anyway) when I was playing with the numbers.

One other bit of structure I was able to notice in the data after re-reading the paper today was a different set of clustering. The triples with three of the same numbers have the lowest counts, triples with two of the same number in a row have (generally) the next lowest counts, triples with two numbers that are the same, but not in a row have (generally) the next lowest counts, and triples with three different numbers have (generally) the highest counts. *I think* their paper predicts this ordering.

So, a really fun week of playing around with prime numbers. There are still a few things to think about – the e-mail from the paper’s authors, and seeing if there’s any way to improve the predictions – but I’m extremely happy with how this little side project went this week. Haven’t had that much fun learning new math in a long time 🙂