Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

Last week we did made cookies using Laura Taalman’s tiling pentagon cookie cutters:

Screen Shot 2016-07-17 at 9.46.03 AM

You can find Taalman’s program here:

and our project is here:

Tiling pentagon cookies

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:

There’s something about pentagons by Evelyn Lamb

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!

Tiling pentagon cookies

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 🙂

10 Pretty to implement math activities for kids

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 🙂

Looking at the new discovery about primes with kids

Earlier in the week Robert J. Lemke Oliver and Kannan Soundararajan of Stanford announced a totally surprising discovery about prime numbers. Evelyn Lamb has a fantastic article about the paper here:

Peculiar Pattern Found in “Random” Prime Numbers

The new paper suggests that the prime numbers are not distributed quite as randomly as mathematicians had previously expected. In particular, the last digit of consecutive prime numbers has a distribution that is different from what you’d expect if the distribution of primes was random.

I thought this would be a fun result to discuss with kids. One surprising thing about this result is that it is pretty easy to understand. In fact, you can double check the result on a computer really quickly.

Before jumping in to the result about primes, though, I wanted to spend a few minutes talking about probability and probability distributions. This part of the project turned out to have some extra fun when my older son asked a neat question about dice.

Here’s the introduction to today’s project and my son’s question – what is the probability of seeing at least one 5 when you roll two dice?

With the complimentary counting problem behind us, we moved on to talk a little bit about the difference between probability and probability distributions. Once again we had a little detour following a statement that my older son made. I was happy to have these extra little conversations about probability this morning:

The next part of the project involved talking about prime numbers. We talked a little bit about how mathemticians viewed the primes. Number theory and prime numbers are not my field – hopefully the details in this part are right. A great (and accessible) read about prime numbers and randomness can be found in Jordan Ellenberg’s How Not to be Wrong.

It was really fun to hear what the kids had to say about the last digit of consecutive prime numbers here. This problem is a great way to get kids thinking about math.

Now we moved to the computer and used a little Mathematica program to study what the distribution of the last digit of consecutive prime numbers looked like. We chose to look at prime numbers that ended in 1. It was neat to see the results. Loved hearing what my younger son observed: “it seems like there’s a lot less primes ending in 1”:

We wrapped up by looping through the first 25,000,000 primes and compared our results to the results that were given in Evelyn Lamb’s write up from above. Our results were really close – yay!

So, I think that talking about this new discovery makes for a really fun project for kids. I’m sure that our project could be improved quite a bit by any mathematician who had a good understanding of number theory (since my understanding of that subject is essentially zero!), but even the high level walk through that we did today was fun. It is pretty amazing to find a new discovery about prime numbers that can be understood by kids!