*Revisiting Richard Evan Schwartz’s Gallery of the Infinite

At the beginning of this year I bought a copy of Richard Evan Schwartz’s Gallery of the Infinite:

We ended up doing a fun project using the book:

Talking through Ricahrd Evan Schwartz’s Gallery of the Infinite with kids

Yesterday I saw the book on our shelf and asked the boys to find something in it that they thought was interesting. The section on bijections caught the eye of my younger son, and we used that section for a project today.

First we talked about the basic idea of bijections and how you could use bijections to tell if two sets were the same size:

Next we talked about a bijection that is pretty challenging for kids to find -> a bijection between the positive integers and the set of all integers:

Finally, we talked through Cantor’s “diagonal argument” which shows that there is no bijection between the integers and the real numbers (and, thus, that the “infinity” of real numbers is somehow larger than the “infinity” of integers!):

Tomorrow we’ll talk through the section of the book that my older son thought was interesting -> the Cantor set.

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A simplified version of the Banach-Tarski paradox for kids

Yesterday we were listening to Patrick Honner’s appearance on the My Favorite Theorem podcast. Honner was discussing Varignonโ€™s Theorem. We actually have discussed this appearance before, but the kids hadn’t listened to the podcast, yet:

Sharing Patrick Honner’s My Favorite Theorem appearance with kids

After listening to the podcast I asked my older son what his favorite theorem was:

However, after giving up on the idea initially (!) I looked at the Wikipedia page for the Banach-Tarski paradox and found an idea that I thought might work. Here’s the page:

Wikipedia’s page on the Banach-Tarski paradox

The idea was to share the first step in the proof – exploring the Cayley graph of F_2 – with kids. Here’s the picture from Wikipedia:

Screen Shot 2018-03-18 at 8.26.06 AM

So, here’s what I did.

First I introduced the boys to some basic ideas about a free group on two generators. I used a Rubik’s cube to both demonstrate the ideas and to show why a Rubik’s cube didn’t quite work for a perfect demonstration (I know that part of the video drags on a bit, but stay with it – there is a nice surprise):

Next we talked about the free group with two generators in more detail. My younger son accidentally came up with a fantastic example that helped clarify how this free group worked.

Then there was a bit of a surprise misconception that I only uncovered by accident. That led to another important clarification.

So, completely by accident, we had a great conversation here.

In the last video they boys thought you could use the “letters” x, x^{-1}, y, y^{-1} only once. In the beginning of this video I clarified the rules.

Next we began to talk about the representation of our free group by the Cayley graph from Wikipedia pictured above. I was really fun to hear how the boys described what they saw in this graph.

Finally, we looked at two different ways to break the Cayley graph into pieces. This video is a little long, but it has a simplified version of the main idea in the Banach-Tarski paradox.

The first decomposition of the Cayley graph is into 5 pieces -> the identity element, words that start with a, words that start with a^{-1}, words that start with b, and words that start with b^{-1}. This decomposition is pretty easy to see in the picture.

The second – and very surprising decomposition is as follows:

The combination of (i) the words that start with a and (ii) a multiplying (on the left) all the words that start with a^{-1} gives the entire set. The same is true for the combination of (i) the words that start with b and (ii) b multiplying (on the left) all the words that start with b^{-1}

Although the words describing this decomposition might not make sense right away, you’ll see that the boys had a few questions about what was going on and eventually were able to see how this second decomposition worked.

And this second decomposition gives a huge surprise -> we’ve taken 4 subsets, combined them in pairs and created two exact copies of the original set. Ta da ๐Ÿ™‚

This project is an incredibly fun one to share with kids. I’m pretty surprised that *any* ideas related to the Banach-Tarski paradox are accessible to kids, but the simple ideas about the Cayley graph of F_2 really are. Using those ideas you can show the main idea behind the sphere paradox without having to dive all the way into rotation groups which I think are a little more abstract and harder to understand.

Anyway, this one was a blast!

Talking though Richard Evan Schwartz’s Gallery of the Infinite with kids

We received Richard Evan Schwartz’s Gallery of the Infinite in the mail this week:

I thought that the boys would love reading the book and asked them to each read it twice prior to today’s project. Here are some of the things that they thought were interesting (ugh, sorry for the focus problems . . . .) :

The first thing the boys wanted to talk about was the “smallest” infinity -> \aleph_0. Here we talked about that infinity and other sets of integers that were the same size.

Next we moved on to talk about the rational numbers – we had a good time talking through the argument that the “size” of the rational numbers was the same as the positive integers.

This argument is represented in the book by a painting of a shark!

Now my older son wanted to talk about Cantor’s diagonal argument. He was a little confused about the arguments presented in the book, but we (hopefully) got things straightened out. I think this shows kids can find ideas about infinity to be really interesting.

Finally, we wrapped up by talking about the implications of the infinity of binary strings being larger than the infinity of counting numbers.

Definitely a fun project. I love the content of the book and so do the kids. The only problem is that the quality of the binding is awful and although we’ve only had the book for a few days, it is falling to pieces. Boo ๐Ÿ˜ฆ

Sharing Kelsey Houston-Edwards’s Axiom of Choice video with kids

Kelsey Houston-Edwards has a new video out about the Axiom of Choice:

The video is amazing (as usual) and I wanted to be able to share it with the boys. This one is a bit hard than usual – the topic is pretty advanced to begin with and is also pretty far outside of my own knowledge – but we gave it a shot.

Here’s what the boys thought after seeing the video:

Next we reviewed how Houston-Edwards divided the numbers from 0 to 1 into buckets. The boys didn’t quite have the details right, but that actually made talking through the idea pretty easy – I learned from their explanation what points needed to be re-emphasized.

Now we talked through the really challenging part of the video -> creating the set with no size. Given the challenge of explaining this idea to kids, I’m pretty happy with how the conversation went here. Also, I only finally understood the argument myself while I was explaining it to them!

Now we backed away from the complexity of the Axiom of Choice and reviewed two other slightly easier ideas that came up in our discussion. Here we discuss why \sqrt{2} is irrational:

Finally, we wrapped up by discussing why the rational numbers are countable:

Although kids will have a hard time understanding all of the ideas that Kelsey Houston-Edwards brings up in her Axiom of Choice video, I think it is fun to see which ideas grab their attention. The idea that you can have a set that doesn’t have a size is pretty amazing. I was pretty happy with how things went today – exploring the ultra complex idea first and then backing off to discuss slightly easier ideas involving infinity. Definitely a fun set of ideas to plant in the minds of younger kids ๐Ÿ™‚

Sharing Kelsey Houston-Edwards’s video about Pi and e with kids

Yesterday I a new video from Kelsey Houston-Edwards that just blew me away. At this point I don’t have the words to describe how much I admire her work. What she is doing to make challenging, high level math both accessible and fun for everyone is amazing.

The new video was about this question on Math Overflow from Erin Carmody:

If I exchange Infinitely many digits of Pi and E are the two resulting numbers transendental?

Before showing the boys Houston-Edwards’s video, I wanted to see what they thought about the question. So, we just dove in:

Next, I took a great warm up idea from Houston-Edwards’s video and asked the boys if they could find *any* two irrational numbers that you could use to swap digits and produce a rational number.

Now, with that little bit of prep work, we watched the new video:

After the video we talked about what we learned. I think just tiny bit of prep work we did really helped the boys get a lot more out of the video.

One of the fun little challenge questions from the video was to show that (assuming \pi and e differ in infinitely many digits, then you will produce uncountably many different numbers by swapping different digits. I didn’t expect that the boys would be able to construct this proof, so I gave them a sketch of how I thought about it (and hopefully my idea was right . . . . )

I think that kids will find the ideas in Houston-Edward’s new video to be fascinating. It is so fun (and sadly so rare) to be able to share ideas that are genuinely interesting to professional mathematicians with kids. As always, I can’t wait for next week’s PBS Infinite series video!

Talking about “The Cat in Numberland”

Last we did a couple of projects based on Kelsey Houston-Edwards’s video about infinity:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Extending our project on Kelsey Houston-Edwards’s infinity video

I got a comment from Allen Knutson on the 2nd project recommending using “The Cat in Numberland” to talk about infinity with kids. I ordered the book immediately and had the boys read it a few times this week. We got around to talking about it this afternoon.

Here’s their initial reaction to the book:

In the last video we I asked the boys for 3 ideas from the book that they wanted to talk about. They chose:

(1) When “Hilbert’s Hotel” is full, how do you fit one more person in?

(2) How about fitting in 26 more people?

(3) When you take away half the people how can the hotel still be full?

Here’s the explanation for part 1 – the idea here shows one strange thing about infinity!

Here’s part 2:

My older son got a little confused by the numbering of the hotel rooms in this video. The numbering of the rooms is hardly the main point, but it is nice to be able to review / revisit some counting ideas in this unusual context:

For part 3 we had a nice conversation about how you can form a bijection between the counting numbers and the non-negative even integers. That conversation went pretty fast so I asked the boys to each find another bijection and got really lucky when they picked two pretty cool ideas – powers of 2 and prime numbers.

The last movie ended with a question about whether or not the primes were infinite. This was also hardly the main point of the project, but turned out to be a fun way to end the conversation today.

So, thanks to Allen Knutson for pointing me to the book and to Kelsey Houston-Edwards for the Infinity video which has now led to three fun projects with the boys!

Extending our project on Kelsey Houston-Edwards’s Infinity video

Yesterday we did a project inspired by Kelsey Houston-Edwards’s latest math video:

Here’s a link to our project:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Last night my younger son and I were talking a little bit more about the project and he asked me why Cantor’s diagonal argument for why the set of real numbers is larger than the set of Natural numbers doesn’t work for rational numbers!! Yes!!

We explored that question today. First we did a quick review of the diagonal argument (which was the last part of yesterday’s project) and then we began talking about the rational numbers:

Next we looked at what would happen if you applied the diagonal argument to the rational numbers:

After getting our arms around the diagonal argument when applied to rational numbers, we backed up and looked at the argument why rational numbers are countable.

Unfortunately I made an easy concept hard in this part of the project. I was trying to explain the “easy for me” idea that if a set that is larger than the rationals was the same size as the natural numbers, that meant the rationals must also be the same size as the natural numbers. My explanation started off terribly and went down hill . . . .

Finally we looked at one of the strangest consequences of all of this infinity stuff. In math language – the rational numbers have measure zero.

The idea here always blows my mind and is a really fun idea about infinity to share with kids.