58

Draw the bottom three-quarters of an oval: Flesh that out to make the bottom half of the strip: Connect one of the open ends at the top to the bottom on the other side: Now draw a straight line across the top: Finally fill in the last edge at the back: An advantage to this approach is that it highlights the fact that the mobius strip starts out as an ...


16

I'm just finishing up a graduate course in computational topology which could be adapted very effectively for this purpose. We're focusing on topological data analysis and computational homology. All the topology in the course has been self-contained, meaning that essentially no previous experience in topology was required. The book we're using is ...


14

Homotopy equivalence v. Homeomorphism. I believe an accessible difference between homotopy equivalence and homeomorphism is that one preserves an intuitive (though hard-to-define) topological invariant, while the other almost never does: the invariant of dimension. For instance, any $\mathbb R^n$ is homotopy equivalent to a point. Similarly, $\mathbb R^n - \...


14

The middle third is obtained by trimming the edges off the original möbius loop. It is therefore simply a thinner möbius loop (the short loop). The outside thirds of the möbius loop are obtained by cutting the loop in half and trimming 1/3 off the edge that was not originally the outside edge. It is the same as cutting the strip in half: as the outside ...


13

You could explore maps such as the London Underground where the actual distance and locations of stops are unimportant only their relationships. Perhaps this could be turned to a practical lesson somehow? Kids with playing the part of stations and holding string? Design a similar map of their home city? Ranking maps? What makes a good map? You could take ...


12

I searched google images and found many nice renditions. Here one that you may prefer from this link (source: umich.edu) . I especially like that it shows the width of the paper and doesn't draw the strip as a line or piece of string. Having the students make them is always a hit.


11

Two ideas. (1) Form a Möbius strip out of paper. And then cut it down the middle. Restart and cut it at a third of its width. See, e.g., "Explanation for cutting a Möbius strip at one-third its width."             (Image from Kidzone.ws.) (2) You already cited the previous question about triangulated surfaces, presumably ...


8

I am not entirely sure on the best way to convey the difference between homotopy equivalence and isomorphic homology groups (or even isomorphic homotopy groups, though on CW-complexes I guess this isn't as big of a concern), except by way of examples. I remember my algebraic topology exam had an explicit example of spaces with all the same homology and ...


8

Wow, thanks for the recent shout-out. I hope this is the right place for me to add a few references that might be useful and haven't already appeared in the answers. Rob Ghrist has just written a fantastic new book called Elementary Applied Topology which provides a soaring and current overview of the field. There are no exercises (yet!) but the figures are ...


8

Since I use Seifert's algorithm to construct surfaces a lot, I tend to go for the flat approach:


8

What you want is Omega Colored Chalk:                  "This non-toxic, low-dust chalk delivers extra-smooth and easily erasable writing in eight vibrant colors." I won't include a link, but if you search, you will find many office-supply outlets that carry it in the US, for about ...


7

Thanks everyone for your tips! I thought I would share what we decided to go for in the end. We had far more material than this (it naturally extends to the investigation into mobius strips, but in an hour this is a good amount of material. Even for top set Year 8 students, who we also did this lesson with in the end). What surface is the game "snake" ...


7

You may find the SageMath knot and links capabilities useful for computation and visualization. The Knot Atlas might be a bit more comprehensive than you are looking for but is certainly a reference to be quite aware of.


7

First of all, let me echo all the comments -- the key point here is that these surfaces are homeomorphic, but this homeomorphism cannot be realized by an isotopy. This is an important distinction! It is probably worth taking some time to explain the issue. In particular, students should understand that topological spaces don't naturally come embedded in $\...


7

If you're going to be an economist, even doing highly theoretical work, then the vast majority of the field of topology is going to be of zero relevance to you. Very little of it has any practical application in the sciences. What we actually use in the sciences is mostly just the topology of manifolds, which is a tiny, tiny piece of the larger picture of ...


7

(1) Here is Margie Hale's tree: (2) And here is Gaspard Sagot's hierarchy:


5

(Moved from comment...) Examples in the plane: a twice-punctured annulus shows the difference between homotopy and homology, because the fundamental group (free on two generators) has a much smaller abelianization. That is, for $s,t$ two loops, one around one hole, the other around the other, $sts^{-1}t^{-1}$ is non-zero in homotopy, but is homologous to $0$....


5

Instead of drawing on the blackboard, use a 3D model and pass it around when you're finished using it for demonstration. If the class is small enough, you can bring in a strip of paper for each student to make his/her own. The demonstration will be much more powerful this way and the lesson more likely to stick.


5

I don't know much about knot theory but I know that Meike Akveld taught knot theory at both high school and university level. Here's a bibliography of one of her courses at ETH Zürich: https://www2.math.ethz.ch/education/bachelor/lectures/fs2015/math/knot/bibliography_FS2015.pdf It includes Englisch and German books both for high school and university ...


4

I recall a cute introduction to the work of Miryam Mirzakhani that is written at the level of a younger student. Something like this could be adapted and incorporated into a 11-year-old's math curriculum. Here is the link to the infographic on Matific. It talks about surfaces and the genus of surfaces and is reasonably well explained. Keep in mind your 11-...


4

While the geometry of surfaces is a relatively quick starting topic it does not seem to me that it is the ideal way to show an 11 year old how much geometric insight can be obtained without metrical information. My favorite example for introducing topological ideas is "Euler's traversability" theorem which says that a graph (diagram of dots and lines) has a "...


4

I have used polydrons with 5th-grade students through to college students:           A store in Massachusetts used to sell them, but recently I've had to purchase them from England.


4

The outer two-thirds are the same piece; the inner third is a different piece from the outer two-thirds. There have to be two different pieces, because at all points along the cut line, the cut is separating a piece that "contains the centerline" from a piece that "contains the original edge". Because both edges are the same edge, the outer two-thirds form ...


4

For a long time, until I transferred to a school with white boards and dry erase markers, I actually used sidewalk chalk like these It is cheap, lasts a long time, incredibly durable, and has nice bright colors. Sidewalk chalk also draws really thick lines making your writing more visible from further away. Additionally, many sidewalk chalk packs come with ...


4

Munkrese is a good book for introductory Topology in part because it has a lot of background at the start. If I recall correctly, the first 70 pages aren't so much Topology as they are just basic abstract math background. You can get a copy pretty cheaply and when you ask questions about it there are lots of people who have also studied it and can help you ...


3

You might try a couple of puzzles near the middle for them to think about, and see if they can come up with the answers. One of my favorites involves a (topological deformation of) two-holed doughnut with a door handle going through both holes, and attached to a door, naturally. The idea is to deform the doughnut so to free one of the holes. (You may be ...


3

Edelsbrunner's new book, A short course in computational geometry and topology. Springer, 2014. (Springer link) might serve as a course skeleton. It is indeed short (110 pages), and written in his laconic but precise style. And beautifully illustrated.      


3

Sphere: No problem. Torus: Swim Ring. Double Torus: Figure Eight (rock climbing), but could be to small for your purposes. $n$-Torus: Craft them yourself fronm Polystyrene and cover them with paper to draw upon. Non-orientable, bounded surfaces: Hmm, not possible, if you really want to draw triangles upon them.


3

How about giving them the five room puzzle on the plane and then on the torus? This is a rather visceral way to appreciate why "holes matter", and may segue nicely into topics in topology.


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