 All right, so what is this? This is a much shortened version of an outreach activity I did for middle school students, but I think it is adaptable to younger and older students. And then the longer slides and the handout from this activity are on my website, or you can email me and I'll send them to you if you're interested. So why? I think one thing that unites a lot of us here at this program is interacting with students, whether through teaching or outreach. So I hope that this is interesting and relevant for a lot of people. So it's fun for our students, but also a fun for us if you've never thought about this thing before. So when I was talking to the students, I started out by telling them a little bit about what topology is. So I had them think about what does it mean for two things to be the same? So an example is color versus color with a U are literally different words, but we consider them to be the same since they correspond to the same entry in the dictionary. Other examples are modular arithmetic and congruent or some similar triangles. So like what I mean by this is our notion of sameness changes depending on the context and what we want to accomplish. Went too fast on that one. All right, so what is topology? Very vaguely, it's a world with its own notion of sameness. So we study shapes and spaces and our notion of sameness is called topologically equivalent. So what do I mean by that? So a sphere, if you haven't, I introduced it to the kids as a soccer ball, so tollo on the inside. Something you're allowed to do to it is bend and stretch. So those pictures are topologically equivalent to the sphere. What you're not allowed to do is poke holes, cut, tape, stuff like that. So those pictures on the right are not topologically equivalent to the sphere, so we would not consider those to be the same thing. So if you've ever heard the joke that a topologist can't tell the difference between a coffee mug and a donut, that's where this comes from. You can take your coffee mug and bend and stretch, we like to say deform all the way to a donut. And I think there's a nice video of this on YouTube, but yeah. All right, another interesting shape that we care about or space is a torus, and so this is an inner tube or the surface of a donut, so it's hollow on the inside. And so a question that I pose to the middle school students is a sphere topologically equivalent to a torus. Probably many of you have thought about this before, but if you haven't, you have about five seconds before the answer will be revealed. No, it's not. And most of the middle school students guess this too, no matter how you bend and stretch the sphere, you can't make a torus. How do you rigorously prove this is maybe not so obvious, and this is the type of thing that topologists are interested in. So how do you rigorously prove that that's true? All right, so then I switched to the game, the torus tic-tac-toe. So here we have a picture of just regular tic-tac-toe. When you win, you can win vertically, like it's shown there, or we can do three in a row horizontally or diagonally. But now I'm gonna change up the game, and so I'm gonna take opposite sides and glue them together or consider them to be the same. So those orange ones are the same. So if you go out the right, then you come in the left, and the top and bottom are the same. If you go up the top, you come in the bottom. And so how does changing the board affect the game? Well, I claim that now this is another winning position besides the three types of positions I mentioned earlier. And so again, you have about two seconds to convince yourself or to think about do you believe that this is true before? Okay, so this is why it's true. That X is on the bottom and kind of right to the left, you can see of that orange line. So we can think of it also as being over there. Those two red X's are the same, and now we have three in a row, which is a valid win in tic-tac-toe. And so at this point, I stopped my lecture and I had a handout for the students, and so I had them pair up and play the game, and they latched on to it really fast, and they came up with notation for how to keep track of what they were doing, and that was really cool. And along with the handout, I had a few questions that I had them think about, which are what are all the ways, the new ways you can win with this rule. And then unlike traditional tic-tac-toe, this game will never end in a draw, and so I challenged them to think about why. So this was like a take home question for them to think about. When I did this activity, I actually didn't tell them the name of the game until the end. So I said it's called Tourist Tic-tac-toe. Does anybody know why? And surprisingly, a few did. And again, if you don't know why, maybe see if you can think about why, but you don't have much time because it's gonna be shown about right now. Okay, so if you actually look at the board and literally identify your sides together, if we identify the orange sides, we get a cylinder, and then we identify our green sides, and then we get a torus if you're convinced by that picture. So, torus tic-tac-toe is really playing tic-tac-toe on a torus. So I can't take credit for this idea. It came from this book, The Shape of Space by Jeffrey Weeks. This is an awesome book and very accessible. So if you found this interesting look there for more. So thanks for listening.