Teaching manifolds to high schoolers

I would like to introduce the concept of a manifold to a high school student.

This person isn't familiar with the axioms of set theory nor topology. However, despite these deficits, I would like to get the general point across that manifolds are useful for studying gravity.

Does anyone know of any resources that address the general idea of a manifold for someone with this level of knowledge?

• Welcome to MESE, Stan! I see you've already accepted an answer. Consider waiting more time before you do that in the future. Giving time for users to answers brings the possibility of greater answers to come along, from which you can choose the best later. – Mark Fantini Dec 5 '14 at 9:56
• Thanks for the advice. Will do going forwards. – Stan Shunpike Dec 5 '14 at 13:00
• savoir-sans-frontieres.com/JPP/telechargeables/… – Carsten S Apr 5 '15 at 13:42
• You have the 'simple' Wikipedia explanation: simple.wikipedia.org/wiki/Manifold – icc97 Feb 12 '18 at 11:47

Introducing the concept of a manifold in general might be hard, but working with specific examples shouldn't be.

The idea that each point $x$ of a n-manifold $M$ has local coordinates in $\mathbb{R}^n$ (for real manifolds) can be realized with the idea that there are projections from parts of a globe ($S^2$) onto the real plane ($\mathbb{R}^2$). This generally shows that (via stereographic projection, which shouldn't be too hard to visually show with a physical globe) the sphere is a 2-manifold. You can even get into the fact that there are many such projections (different coordinate systems on the manifold) by using different mapping types (the wiki page has many discussions on the subject) and even get into the fact that $S^2$ is not $\mathbb{R}^2$ by showing where several types of projections fail and distort.

You can then diverge into discussions of other common manifolds (ones embedded in $\mathbb{R}^3$ would be best so they can visualize them) such as the Torus, the Mobius band, or even lines.

That should cover the concept of "what is a manifold" with enough of a concept image that they might appreciate some of the idea of how they interact in the study of gravity. But alas, I am unfamiliar with the subject myself to give voice to that specific example. I would be unsurprised that it would be a challenge to give them any specific mathematics for it besides a "gut feeling" that manifolds are involved. Perhaps using rays of light as geodesics in this coordinate system on the gravity wells would be useful.

• That's clever. Thanks. I wanted to at a minimum get across that not all "spaces" have the simple, linear structure of Euclidean spaces. The examples you gave should at least get that point across nicely. – Stan Shunpike Dec 5 '14 at 8:07

If you try to convey the idea of what a manifold is, and if you do not want to restrict to surfaces (and curves), then at some point you will have to explain what an intrinsic manifold is (as opposed to a submanifold). Many mathematicians working in areas remote from differential geometry are happy with working with submanifolds when they need to, but they are familiar with $\mathbb{R}^n$, $n>3$; a general audience will not fall for that trick.

Now, how to explain intrinsic manifold to an audience with less mathematical knowledge than our respected colleagues? Chris C already suggested using the sphere as an example; I second him on this, except that:

1. rather than stereographic projection, I would use longitude and latitude to show that a position on earth can be defined by two numbers, as it relates to known concepts;

2. distortion of maps is a great subject, but it is a metric property, not a differential-topological one, so I would avoid it in a first approach,

3. one can use earth to explain intrinsic manifold in a convincing way.

Below, I will first give details about 3. above, and after that I'll give a way to deal with higher dimensions (at least, with dimension 3).

Instead of having your audience vizualizing Earth inside the Euclidean space, let them instead imagine people in very primitive time, before the roundness of earth was known, who are only aware of some relatively small region around them (hey! here is a chart! but don't tell them yet). Clearly, first neglecting hills, mountains and valleys, one would easily consider standing on a Euclidean plane, even without knowing formal Euclidean geometry. It is probably better to avoid discussing too much geometry, as it would lead us far from the core subject; but the following "obvious" property is useful to discuss: if one starts walking straight ahead and never stops, then one gets farther and farther away from its starting point, forever.

Now, we can appeal to the higher knowledge we share with our audience: the Earth is in fact a sphere! For the sake of the argument, let us forget about slight geometrical distortions as Himalaya and slight inconveniences as oceans, and let's assume Earth is a perfect round sphere. Then the obvious property above is of course wrong: if one walks straight for a sufficiently long time (80 days?), then one gets to its starting point again after a whole turn around the world. Here you are starting to touch the main point: there can be "spaces" (i.e. set of positions one can occupy) where locally everything looks like a plane, but that have different global properties. These spaces are exactly manifolds. Note that the neglected inconveniences above can be taken into account without harming the point too much.

At this point, it is certainly a good idea to propose another example; the torus is the obvious one, but instead of talking about a donut, let's use the classical video game world: a rectangle where moving up at the upper edge makes you beam out at the bottom, and moving right at the right edge makes you bam out at the left, and vice-versa. This is well known to most, and is more intrinsic than the donut surface. Also, you can use the case when one does not see the whole world, but only the small piece that fits the screen, and where there is no feeling of beaming out at any point: one just go straight up and ends at one's starting point after some time. The edges and the beam stuff are really only a convenience to represent the space on the map screen.

Ok, now it is pretty straightforward to engage your audience with a three-dimensional manifold: simply picture a huge but finite box in space, enclosing Earth, Sun, etc., such that moving through the upper face makes you beam out to the bottom face, etc. But of course, this edges and beam stuff are again only convenient ways to picture this space inside the one we are accustomed with; you have the three-dimensional torus. At this point you can say that more generally, a manifold is simply the global shape of a space (or surface) which is locally just as plain, boring old space (or plane); you could have said that at the very beginning, but it would not have been clear that other examples than plain, boring old space can exist.

If you draw a higher-genus surface in $\mathbb{R}^3$ (these ones are difficult to picture intrinsically), you can easily convince your audience that a lot of different surfaces exist. But after the previous discussion, this leads you to let them imagine that the same should be true in 3D. They may very well be living in a universe that is not shaped as plain, boring old space! Maybe that a rocket sent in a given straight direction would come back to its starting point after some (very long) time!

It also gives you the opportunity to solve a longstanding dilemma. It is difficult to imagine that the universe is infinite, because infinite is difficult to grasp; but on the other hand, it is difficult to imagine that the universe has a boundary, because what would then be on the other side of the boundary? The above discussion, where all examples are closed manifolds, shows that the universe could in principle be finite and without boundary. Making ma able to truly understand that is probably one of the most precious gifts mathematics has given to me.

The next step would be to engage with higher-dimensional spaces and manifolds, but this answer is really long already. I will therefore only point to a reference, which is tremendously clever to explain this: "flatland" by Edwin Abbott Abbott. I vaguely remember that his idea has also been reused in more modern texts.

• Awesome. That is an exceptionally straightforward and vivid explanation of exactly the basic concepts I had hope to convey. If only mathematician's wrote their books that way! I gave this young man the introductory chapter to John Lee's Introduction to Topological Manifolds hoping it might click, but it became clear I needed to seek a different approach to teaching him in more concrete terms...such as these. – Stan Shunpike Dec 7 '14 at 5:14

To obtain some heuristic idea about geometry and physics there are many entry-level, mostly non-technical, books. Many of these books are written to popularize string theory for the masses. Whatever your opinion on that matter, the fact remains that geometrically there is something to be gained from reading classics like Hyperspace by Michio Kaku or Elegant Universe by Brian Greene. Both of these make an effort to explain the interplay between General Relativity and the geometry of manifolds.

In addition, if they are really, really interested, they could learn something from Susskind: Leonard Susskind's General Relativity Lectures

• Lee Smolin's books might be of interest as well. – Chris C Dec 5 '14 at 15:45
• @Chris C Indeed, to be fair, I probably should add a dozen more... – James S. Cook Dec 5 '14 at 16:56
• I'm not sure how much this answers the question, but that is an amazing set of Susskind lectures which I didn't know existed. – icc97 Feb 12 '18 at 11:46

An accessible book that I found inspiring as a HS student was Geometry, Relativity and the Fourth Dimension.

If your student is especially interested in ideas in relativity, then the geometric concepts here are key, without getting bogged down in the technicalities of defining manifolds more generally.