Last night I was watching the Olympics and flipping through one of my favorite recreational math books – “The Penguin Dictionary of Curious and Interesting Geometry” by David Wells.

One of the sections of the book discusses the Schwartz Lantern, and that discussion gave me the idea to talk through this interesting shape with the boys today. Unfortunately that talk didn’t go as well as I’d hoped – too many details are over their head – but, despite my failed attempt, I do think there are some really neat things that kids could learn from the Schwarz Lantern. Students with a little bit of background in geometry and trigonometry will, I think, have a ton of fun (and learn a lot, too!) studying this topic.

The first time I heard about the Schwartz Lantern was reading this wonderful article by Evelyn Lamb last fall:

http://blogs.scientificamerican.com/roots-of-unity/2013/11/30/counterexamples-in-origami/

Lamb has a amazing gift for explaining mathematics to broad audience and this article of hers is one of my favorites. I wish I had the origami skills to make of the lanterns.

After reading Lamb’s piece I did a little more searching and found a nice write up on Cut the Knot:

http://www.cut-the-knot.org/Curriculum/Calculus/SchwarzLantern.shtml

as well as a neat example from the Wolfram Demonstrations Project:

http://demonstrations.wolfram.com/CylinderAreaParadox/

With Evelyn Lamb, Alexander Bogomolny, and Wolfram already covering the topic, only a fool would think he could add something to the discussion . . . so here’s what I did:

The thing that made me stop and think about the Schwartz lantern last night was this curious statement from the Penguin Dictionary – when you approximate the surface of the cylinder using Schwartz’s idea “instead of approximating more and more closely to the surface of the cylinder, the triangles turn against the surface, and the total surface area tends to infinity.”

I honestly didn’t get the point the book was trying to make. As I was sort of day dreaming about the approximation while watching the Olympic last night, I was not finding that the area of the approximation went to infinity. Finally I took out a pencil and paper and figured out where I’d gone wrong. Finding the error that I’d made helped me get a better understanding of why the Schwartz Lantern was so interesting. I thought the kids would enjoy learning about this neat shape (particularly after the discussion of fractals that I had with my younger son yesterday) and decided to try to talk through it with them this morning.

The first thing I did was show them some of the information from the three web pages I linked above. Then we read about the problem in the geometry book, and even drew out an approximation on an empty can. Seemed like as good a way as any to get started:

Next we moved to the whiteboard to draw some pictures and get our arms around the problem. The first thing we needed to do was to count the number of triangles in the approximation. I chose to emphasize this step because it is one of the few pieces of this project that they could do on their own. Next I showed them how to find the surface area of a cylinder so that we could later compare the area we get from the triangles to the actual surface area of the cylinder.

With that mistake out of the way we moved on to the correct calculation of the area of the triangle. This is the part I really wish I’d done better. We got caught up in too many of the details and I feel that the main point was obscured. Nonetheless, we did arrive at the conclusion that as the number of triangles increases, the area of the approximation goes to infinity.

Despite feeling that I could have done this activity a little better, I think that there are some fantastic ideas for kids involving the Schwarz Lantern. It really is amazing to me that you can approximate the cylinder with triangles and get (i) the circumference right, (ii) the volume right, (iii) and miss the area by quite a bit! It shows that you can’t just blindly apply the ideas you learn in calculus to any situation.

But you don’t really need calculus to understand this example – just a little geometry and basic trig. For kids with that background, an hour or so working through some of the geometry followed by another hour of playing around with the origami in Evelyn Lamb’s post would, I think, make for a really fun day (or two) of math. Even though this definitely didn’t go as well as I’d hoped, it was still a fun project for us to work through.

And here is the original source from Schwarz’s Collected Works:

https://archive.org/stream/gesammeltemathem02schwuoft#page/310/mode/2up

Wow – that’s really cool. Thank you.

At first I thought the Schwarz lantern is a fractal, but doesn’t have this nested structure to it, which is typical of fractals. It just is an example of a nonconvergent series. Here is another little gem by Schwarz: the proof that the orthic triangle has the minimum perimeter of all triangles inscribed into an acute angled triangle.

https://archive.org/stream/gesammeltemathem02schwuoft#page/344/mode/2up

I first saw that proof in Coxeter and Lreitzer’s “Geometry Revisited.” It is in section 4.5 and, indeed, attributed to Schwarz.

Thanks for pointing out the connection!