Surfaces are complicated. Triangles are simple. That's an idea behind some methods of creating computer graphics and some advanced mathematics. If we have a surface, we can take a bunch of points on the surface and connect them into triangles to obtain an approximation of the surface. That's all well and good, but how reliable is the triangulation? How accurately does it reflect the properties of the original surface? For example, as we increase the number of triangles in the approximation of the surface, will the surface area of the triangulated surface get close to the surface area of the original surface? Karl Hermann Amandus Schwarz, who first described the Schwarz lantern. Image: public domain, via Wikimedia Commons.

In 1880, mathematician and righteous facial hair maintainer Hermann Schwarz answered this question in the negative by producing a counterexample, a surface and sequence of triangulated approximations for which the surface area of the triangulations gets arbitrarily large and hence doesn't converge to the surface area of the original surface.

Earlier this semester, I had the opportunity to go to an origami workshop put on by the local chapter of the Association for Women in Mathematics. Radhika Gupta, a graduate student here at the University of Utah, showed us how to make this counterexample, dubbed the Schwarz lantern, just by folding paper.

To make a Schwarz lantern, we start with a piece of paper.

We divide the length of the paper into M parts, so we get M skinny horizontal rectangles.

Then each skinny horizontal rectangle gets divided into 2N triangles by placing N points on both the top and bottom at an offset and connecting the points up into triangles. (Note that in the end, the left and right sides will be glued together, so some triangles are drawn so half is on the right side and half is on the left side before gluing.)