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My 12-year-old got interested on topology puzzles and thought it would make a great presentation for his school assignment. I myself am not familiar with topology so am of little help. Basically, he wants to do an introduction to topology that he and his age group could easily grasp and appreciate. Would you have suggestions on what topics and puzzles he can include in his 20-minute presentation? So far, we're thinking he can introduce puzzles in the beginning, make the introduction to topology (definition, genus, etc.) with some slides, explain some real-life applications of topology, and then

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    $\begingroup$ I also want to ask: is the cutting of mobius strips to create a new shapes considered part of topology? $\endgroup$ – Yayeen Gavieres Oct 26 '17 at 11:13
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    $\begingroup$ amazon.com/First-Concepts-Topology-Mathematical-Library/dp/… $\endgroup$ – Paracosmiste Oct 26 '17 at 11:34
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    $\begingroup$ @YayeenGavieres Yes, "cut and paste" techniques are definitely a part of topology. $\endgroup$ – Adam Oct 26 '17 at 12:22
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    $\begingroup$ So probably "an introduction to topology" is beyond a middle-school presentation. But Möbius bands, describing what it means to be one-sided, cutting at the middle line at the 1/3 line would be enough for the whole presentation. $\endgroup$ – Gerald Edgar Oct 26 '17 at 13:27
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    $\begingroup$ obligatory: georgehart.com/bagel/bagel.html $\endgroup$ – James S. Cook Oct 26 '17 at 18:13
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One thing that your son might consider talking about is the winding number. One example that I sometimes give involves garden hoses. When the hose is nicely coiled on the ground and you try to drag it (staying along the ground) to the garden, you develop kinks in the hose. One way of interpreting this is that the winding number is being preserved.

Of course, there are ways of unkinking the hose, but they involve 3 dimensional movement. (In particular that loops on the 2-sphere can be contracted --- $\pi_1(S^2)=0$ --- unlike the case of the unit circle.)

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  • $\begingroup$ Thank you! Will check out the winding number and the garden hose analogy. $\endgroup$ – Yayeen Gavieres Oct 27 '17 at 13:53
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I find the "Topology for Beginners" youtube videos by Richard Southwell easy enough for middle-school-age kids to understand.

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Keep it light and visual (not algebraic) and very, very basic. Moebius strip. Klein bottle. Coffee cup to donut. 7 bridges of Konigsberg.

P.s. It would help to advise you to know how long the talk is.

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  • $\begingroup$ Minimum 20 minutes, but maybe can go up to half an hour. Very high level -- puzzles (like you suggested), basic definition, how shapes are topologically equivalent, and maybe mention some real-life applications (could use more help on this last item). Already ordered an ACME Klein bottle. Thank you! $\endgroup$ – Yayeen Gavieres Nov 1 '17 at 10:44
  • $\begingroup$ Thanks, man. You actually can't cover everything in that amount of time. If you can have the boy bring supplies to school and walk the class through Moebius making and cutting, that would be great. (We are physical beings, learn with hands.) $\endgroup$ – guest Nov 2 '17 at 20:50
  • $\begingroup$ Google search showed low real applications of topology (not a parabola with artillery or trig for surveying). Pure mathers were funny, trying to say that creating THEORETICAL underpinnings of other math were applications. This is not making money, feeding kids, or killing enemies. Pass on the applications--if you have to cover the import of the topic, you could discuss honestly that it is theoretical, fun. For an older audience, show where it fits into a course of study for a math major or how it relates (without math, conceptually) to other subjects is good. But these kids are too young. $\endgroup$ – guest Nov 2 '17 at 21:02
  • $\begingroup$ Got it. Will definitely put in a number of puzzles and tricks with the Mobius strip, etc., and stick to the basics with the discussion. Will probably have to prioritize which to include in the presentation because everyone says we can't cover a lot in just 20 mins. I'm sure my son will want to focus on the fun part. $\endgroup$ – Yayeen Gavieres Nov 3 '17 at 10:33
  • $\begingroup$ Yeah, with 20 minutes, you need to be selective. Have the kid write out what he is going to say and then practice it once or twice. (I am not a natural speaker, but any time I do this, I rock the presentation.) Note: this is a little more work than just making an outline and winging the talk. $\endgroup$ – guest Nov 3 '17 at 15:34

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