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I'm going to give a class to highschoolers about topology. I've prepared the beginning of the class where I introduce what is topology and give them different ways we use to describe spaces, but everything is described very intuitively since they are in highschool. For example, I describe compactness as can we put this space into a ball or dimension as the number of different direction we can go,...

Now I want to tell them some weird and fun fact about topology, and maybe make them do some intuitive exercises, like that the Klein bottle has no inside or outside or the seven bridges of Königsberg problem.

So my question is do you know any topological problem or fun fact that highschooler can understand and find interesting enough that they would want to do topology ?

Edit I found a great book and wanted to share it here: intuitive topology by V.V. Prasolov

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    $\begingroup$ Topology fun fact: some people manage to get a permanent academic position by studying topology! :-) $\endgroup$
    – Asaf Karagila
    Dec 18, 2023 at 9:44
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    $\begingroup$ Make a mobius band and draw on it with a pen. Cut the band along the middle length to create two bands. $\endgroup$
    – Paul
    Dec 18, 2023 at 10:11
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    $\begingroup$ Relevant: Algebraic topology in high school? $\endgroup$ Dec 18, 2023 at 14:06
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    $\begingroup$ The hairy ball Theorem is great for this, because they can all go off and try to come up with a combing of their own. Even more fun when you reveal that it can be done with less than 2 singularities (although I wouldn't call that last point purely topological). $\endgroup$
    – JonathanZ
    Dec 18, 2023 at 22:58
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    $\begingroup$ Fixed point theorem. Take two maps of their home state (or country). Fold, twist and whatnot the other (but do not tear it), and toss it randomly on top of the other. There exists at least one location in the state whose versions appear on the same spot (=on top of each other). $\endgroup$ Dec 20, 2023 at 13:29

11 Answers 11

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Cutting a Möbius strip in half:

J.M.

or at one-third:

MobiusThird

I would bring a collection of ~20 scissors and tape dispensers, and have the students form Möbius strips (sharing equipment), predict what will happen, and then cut the strips.

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  • $\begingroup$ I see the half-cut also suggested in a comment by user Paul $\endgroup$ Dec 20, 2023 at 13:26
  • $\begingroup$ What is the boundary between the two behaviors? We can't precisely cut at 1/2 or 1/3. But that the different phenomena are seen implies that there is some range for each. And then how sharp is the change in behavior and what does it look like if you try to cut at exactly the border (say 5/12 if that is the dividing line) of the phenomena? Also what happens between 0.0 and 1/3? Is that all "like "1/3" in result. [Apologies, but I have tried Googling answer!] $\endgroup$
    – guest
    Jan 11 at 15:35
  • $\begingroup$ @guest: See the figure here to understand what $1/3$-rd means. The photo above was my own experiment. $\endgroup$ Jan 11 at 21:10
  • $\begingroup$ That doesn't answer my questions. Also, you'll see in 2019, I had the same questions. $\endgroup$
    – guest
    Jan 11 at 21:48
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    $\begingroup$ I've done this with high school students and it works. $\endgroup$
    – Dan Fox
    Jan 26 at 9:47
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Not so much a "topological fact" but more of a hands-on exploration with two topological objects:

Eugenia Cheng suggests cutting a bagel (torus) along a Mobius strip to increase the available surface area for cream cheese!

This link describes the method and has a video demonstration.

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This can be presented in ordinary terminology to high school students and even lower. I made it to first and second year college students. While I think a student can appreciate the connection to maxima, minima, and saddle points better if they have had multivariable calculus, I don't think it is necessary. Only some in my audience had had it. Pretty much everyone had seen a donut before.


When you dunk a donut in coffee (or cider), the wet surface morphs. The surface transitions between homeomorphically distinct objects when the donut passes a critical value of a certain distance function: the depth the donut has been dunked.

There are three types of transitions that can be characterized by adding one of three elements, whose boundary is connected to the current boundary:

  • a point (0D)
  • a line segment (1D)
  • a disk (2D)

Once added, the element can spread out and form a (bigger) surface.

For instance, as the donut approaches the coffee, no part is wet and the wet "surface" is the "empty set." As the donut makes contact with the coffee, a point gets wet which spreads out in a bigger wet surface as the donut is dunked deeper. The boundary remains a loop until the next transition.

The next two transitions are a bit tricky to describe. They occur when the donut hole first reaches the surface of the coffee and when it first becomes completely submerged. In between the transitions the boundary consists of two loops.

What one sees at the second critical value (depth = lower inside of the donut hole) is the (single) boundary loop pinching together at a point and immediately separating into two loops. This is not what happens if you connect a point to the single loop boundary. Connecting a point changes nothing. However, it is what happens when you add a line segment, which immediately spreads out. Topologically, this is the correct construction. Unfortunately, most people, even high-schoolers, do not have the appetite to experiment enough times to convince themselves that the construction is good description of what they see happening.

At the third critical value (depth = upper inside of the donut hole), a line connects the two boundary loops and spreads out so that the boundary is again a single loop.

At the fourth critical value (depth = complete submersion), a disk is added. Again this may be hard to see. The boundary appears to "shrink to a point" as the donut goes under. However, at no depth is the boundary at the top a point. Until the top is level with the surface of the coffee, the boundary is a loop. When level, the whole surface of the donut is included. The only way to close the hole is to connect the boundary of a disk to the boundary of the wet surface.

A theoretical connection is that the dimension of the element that is added is equal to the index of the critical value. At each critical point, one of three things happen:

  • a cup: the principal curvatures are up (all sections by a vertical plane are concave up)
  • a saddle: one of the principal curvatures is up, the other down (some sections by a vertical plane are concave up, some concave down)
  • a cap: the principal curvatures are down (all sections by a vertical plane are concave down)

The index is equal to the number of principal curvatures that are down. The transitions have indices $0,1,1,2$ in order (hence point, line, line, disk). That in turn implies an Euler characteristic of $\chi=(-1)^0+(-1)^1+(-1)^1+(-1)^2 = 0$, should you want to make a connection there.

Finally, it's an argument to get your department to spring for donuts for your class.


I believe I got the idea from Milnor's Morse Theory, but I'm on vacation and cannot check.

Marston Morse himself gives a demo (of a different surface, not a donut) which I was shown in college: https://www.youtube.com/watch?v=QVMnv-KpQlM

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    $\begingroup$ Just in case someone wants to read the background maths, what you are describing is a Morse function and the idea of studying the topology of a manifold via Morse functions is generally called Morse theory, en.wikipedia.org/wiki/Morse_theory $\endgroup$
    – quarague
    Dec 21, 2023 at 16:06
  • $\begingroup$ @quarague Thanks. I had actually provided a reference (Milnor, one of the best writers of mathematics). $\endgroup$
    – user1815
    Dec 25, 2023 at 6:02
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The organization Mathematical Puzzle Programs organizes puzzle events aimed at connecting high school students with contemporary/research mathematics.

This poster includes three puzzles, two of which are connected to topology (one boils down to the concept of homeomorphism, the other is one dimensional persistent homology). https://www.mappmath.org/poster

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I gave a well received talk about the topology of spaces of planar polygons with specified side lengths, considered up to rotation. E.g. the space of triangles is empty or a point, of quadrilaterals is usually a circle, but of nearly regular pentagons starts to get you surfaces of various genera.

The main point I was hoping to stress was that mathematicians routinely mix levels - what is a polygon here is a point there.

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    $\begingroup$ Do you have some source or reference to share? You got me interested. $\endgroup$ Dec 23, 2023 at 21:18
  • $\begingroup$ @MichałMiśkiewicz: Here's one source. Shimamoto, Don, and Catherine Vanderwaart. "Spaces of polygons in the plane and Morse theory." The American Mathematical Monthly 112, no. 4 (2005): 289-310. DOI. $\endgroup$ Dec 24, 2023 at 13:11
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Speaking from a high school student's perspective who has attended a few talks about topology before, I think it was really interactive and fun when we created a Mobius strip and attempted to cut it in half like how Joseph O'Rourke mentioned in his answer. One of my teachers also brought a Klein Bottle in so that we could touch it and attempt to trace the path around it, this was helpful since visualising it from 3D images was trickier, however this depends on whether you have a Klein Bottle at home or not... To end the talk, you could recommend a few books for anyone interested in topology, for example, one of my teachers recommended the book 'The Shape of Space' by Jeff R Weeks.

Some of the answers on here are addressing more advanced topics and whilst I think this is great, it seems like OP would want ideas which cater to introductory topics.

Hope the above helps and good luck with your presentation! :)

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Here are a few ideas for exercises that hopefully lead to "fun" facts in (algebraic) topology. If I remember correctly, most (but not all) of them are contained in Prasolov's book you mentioned.

  1. a) Let them try to solve the three utilities problem. This boils down to embedding the graph $K_{3,3}$ in the plane, which is impossible. b) Explain that this is impossible. If you take Euler's theorem (about the Euler characteristic) for granted, the proof is quite accessible. c) Let them try to solve the same problem on the surface of a mug, which is a torus. I'd expect them to succeed with some trial and error, and possibly some hints (the handle lets you avoid one intersection). d) Discuss the difference between the two cases, and how Euler characteristic works for different surfaces.

  2. Let them discover the Borromean rings by themselves. That is, give them three loops (that can be open and closed at will) and tell them to arrange it so that the three loops together are tangled, but removing one loop makes the other two untangled. The solution is not unique, but there's a good chance they will arrive at this one, because it's the simplest.

  3. a) Consider the following puzzle. There are two nails in the wall, and the goal is to hang a picture on them (using a loop attached to the picture), according to two rules: the picture hangs if both nails are in place, but if either nail is removed, it falls down. b) After they arrive at a solution, discuss why it's the same puzzle as the Borromean rings. In both cases we're studying the same fundamental group: of the space minus two untangled circles (in 2) or of the plane minus two points (in 3).

  4. a) Let $A,B,C$ be Borromean rings. Find a surface spanned by $C$ (i.e. a surface with $C$ as its boundary) which doesn't intersect $A$ or $B$. b) Discuss why this surface cannot be a disk (because you could shrink $C$ and remove it without touching $A$ or $B$). c) If possible, introduce Hurewicz's theorem. It explains the difference between spanning a disk (i.e. being trivial in the fundamental group) and spanning a surface (i.e. being trivial in the first homology group).

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Sphere eversion is a good one for several reasons. First, it's very counter-intuitive, and yet you can show them this beautiful animation of it. Second, it's a good way to introduce a lot of important higher-level concepts -- what does it mean to deform something without cutting or tearing it and how does that relate to things like continuity or differentiability? Finally, at a personal level, the characters involved in the story could have walked out of a movie. Stephen Smale gave a non-constructive proof of eversion that plenty of people thought was just wrong at first. There's the famous story about him showing up to a lecture late and copying down a list of problems the instructor wrote down on the board thinking they were homework and solving a couple of them, not realizing that those were unsolved problems in the field. Then there's Bernard Morin, who gave a proof of sphere eversion by constructing the sequence of transformations; Morin went blind at the age of 6. At that rate, you can even talk about the differences between constructive and non-constructive proof.

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I have given talks to high school students about topology previously. I often talk about the surfaces that can be obtained by identifying sides of a square in different ways - this is quite accessible and there are nice animations that show how to obtain the immersion of the Klein bottle in $\mathbb{R}^3$, beginning with a square with edges identified appropriately.

From there, you could go on to talk about 3-manifolds such as the 3-torus and other manifolds obtained by identifying faces in a cube. You could also show them how to play tic-tac-toe on a torus, Klein bottle or projective plane, and how there are new ways to win as a result of the gluing. Jeff Weeks has some relevant games on his website. You could also look at gluings of other polygons, such as gluings of an octagon that give a double torus.

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I like Dirac's string trick.

It shows that a full rotation doesn't bring an object back to its initial state, a second rotation is required. Any object has some kind of invisible parity such that a full rotation in any direction flips the parity.

http://www.av8n.com/physics/dirac-string-trick.htm

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    $\begingroup$ I also think Dirac's string (or belt) trick is a nice example, but you didn't mention topology in your answer. In fact, the trick boils down to the fact that $\operatorname{SO}(3)$ (the space of 3d rotations) has nontrivial loops (and in fact its fundamental group is $\mathbb{Z}_2$). $\endgroup$ Dec 22, 2023 at 21:17
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Proof

Let S be a square with bottom left point at (0,0) and top right point at (1,1)

Let (0,y) be equivalent to (1,y) for any y

Let (x,0) be equivalent to (x,1) for any x

Thus, Pacman is played on the surface of a donut, where Pacman is a member of the set of games which high schoolers don't play these days.

QED

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  • $\begingroup$ When giving a popular lecture on this topic recently, I was surprised to learn how many high schoolers knew Snake. Of course I played it in front of them anyway (in an online Nokia-Snake emulator). $\endgroup$ Dec 21, 2023 at 23:23

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