The Schwartz lantern

It’s thanksgiving, let’s have some fun in lantern-making!

This thing called Schwartz lantern initially came up in a talk some years ago, I vaguely remembered it as ‘a cool example where the refining triangle approximation of a smooth surface fail to converge in area’. Anyways, it came up again recently as I was talking to some German postdoc working in ‘discrete differential geometry’. As the example was mentioned, he took a napkin and started folding…and I just realized this lantern can actually be made with a piece of flat paper!

Given a compact, smooth surface (possibly with boundary) embedded in \mathbb{R}^3, we can approximate it by a PL surface with all vertices on the surface and having only triangular faces. (just like what’s done in many computer graphic softwares nowadays).

Question: When the vertices of the faces gets denser and denser and the diameter of triangles converge to 0, does the area of the PL surface converge to the area of the surface?

Ok, to explain why I found this being a quite curious little question, let’s prove a couple of trivial observations:

Trivial fact #1: The sequence of PL surfaces as described above does Hausdorff converge to the smooth surface.

Proof:Since our surface is smooth, as the diameter of the triangles converge to 0, the length of the geodesic between two vertices is roughly the length of the edge in the 1-skeleton of the PL surface. In particular, less than twice its length.

Now by compactness we have a positive injectivity radius, implying that for small enough length c, the diameter (on the surface) of region enclosed by any loop of length most c is controlled by c (say it’s \leq f(c) which converge to 0 as c \rightarrow 0). Now the \mathbb{R}^3 diameter of the region is of course even smaller than its surface diameter.

In conclusion, when all triangles have small diameter \varepsilon (hence all its sides have length at most \varepsilon), the geodesic triangle on the surface has parameter less than 6 \varepsilon . So \mathbb{R}^3 diameter of the geodesic triangle is no more than f(6 \varepsilon). Hence the surface is contained in the f(6\varepsilon)-neighbourhood of the vertex set.

Obviously the PL surface is also contained in this neighbourhood. Hence the Hausdorff distance is at most f(6\varepsilon), which converges to 0.

Trivial fact #2: For curves in \mathbb{R}^2 (in fact, or \mathbb{R}^n), the length converges.

Proof: Well…what can I say…see any undergrad calculus book? (well, all we need is that smooth curves are rectifiable. Of course they are…

So from the above observations, does it kinda smell like the area would converge? (If you know the answer, you should pretend you don’t and nod at this point :-P) Well, the fact is they don’t have to converge! (otherwise why are we making counter-examples here?) Furthermore, this is first discovered by a super-cool dude – Schwartz! He even wrote a paper about it back in 1880.

How can this be possible? You might have already observed that with some simple curvature bounding, we can push the argument for trivial fact #1 to show that area of the curved surface is controlled above by the straight surface, The point being (perhaps to one’s surprise) that the ‘straight surface’ can be a lot LARGER than the curved one!

So the example is a sequence of ‘lanterns’ converging to a standard cylinder (say of height and circumference both equal to 1), i.e. PL surfaces with triangular faces, vertices on the cylinder, with smaller and smaller ‘grids’, and the sum of areas of the triangle blows up to infinity.

As shown above, if we have put N^4 points on the cylinder, N points along the circumference and N^3 in the vertical direction (picture is not to scale); connected to form triangles in the above way.

Now all triangles are isosceles and identical. Doing some middle-school geometry shows that they have base length \geq \frac{1}{2N} and height \geq \frac{1}{4N}\tan^{-1}(\frac{\pi}{2N}) (this calculates the distance between the midpoint of the base to the cylinder surface).

Having 2N^4 triangles means the area A_N of the PL surface is at least N^4 \times \frac{1}{2N} \times  \frac{1}{4N}\tan^{-1}(\frac{\pi}{2N}) when N large, \tan^{-1}(\frac{\pi}{2N}) \sim \frac{\pi}{2N} \geq \frac{1}{N}. Hence the A_N \geq \frac{N}{8} blows up to infinity.

Is this pretty cool? This lantern also have an interesting feature that, if we define ‘curvature’ on vertices to be the sum of angles attached to that vertex, (and of course the curvature on the edge between two flat faces shall be 0), then all lanterns have curvature 0 everywhere, just as in the smooth cylinder! i.e. it can be made by folding a single piece of flat paper.

Let’s note that although the triangles are getting uniformly smaller, they do become ‘thinner and thinner’ in the example. In fact this is the only way it can go wrong, i.e. it can be shown that if we further require the triangles to have bounded eccentricity then the area does converge.

Add-on: I actually made the lantern! They are interesting to fold, aesthetically pleasing and even functional! (you’ll see light flaring out in an interesting way)

Trying it out while one thinks about problems is highly amusing and recommended~

All one needs to do is:

Tips on folding:

1. Be sure to make all diagonal lines positive fold and horizontal lines negative.

2. Make the diagonals cross an even number of horizontals or else after you finish all diagonals, you’ll end up with left and right-facing diagonals not crossing on the horizontal (i.e. you’ll need to double the number of horizontals to make it work again)

3. After finishing all lines, it might be hard at first to make the whole thing ‘fold up’. The trick being to make sure all ‘crosses’ are ‘poped-out’ on the whole surface. The final folding process does not work locally!

4. Although theoretically you can take an arbitrarily long strip of paper with unit width to make unit-sized lantern, but in order to not make a million folds and have super-sharp angles between the diagonal and horizontal; I recommend not being too aggressive on the length :-P (square-ish papers are good enough)

Have fun!~

A not-very-good picture of my lantern (larking light bulb…>.<)

A report from the Workshop in Geometric Topology @ Utah (part 1)

I went to Park City this passed week for the Workshop in Geometric Topology. It was a quite cool place filled with ski-equipment stores, Christmas souvenir shops, galleries and little wooden houses for family winter vacations. Well, as you may have guessed, the place would look very interesting in summer. :-P

As the ‘principal speaker’, Professor Gabai gave three consecutive lectures on his ending lamination space paper (this paper was also mentioned in my last post). I would like to sketch some little pieces of ideas presented in perhaps couple of posts.

Classification of simple closed curves on surfaces

Let S_{g,p} denote the (hyperbolic) surface of genus g and p punchers. There is a unique geodesic loop in each homotopy class. However, given a geodesic loop drew on the surface, how would you describe it to a friend over telephone?

Here we wish to find a canonical way to describe homotopy classes of curves on surfaces. This classical result was originally due to Dehn (unpublished), but discovered independently by Thurston in 1976. For simplicity let’s assume for now that S is a closed surface of genus g.

Fix pants decomposition \mathcal{T} of S, \mathcal{T} = \{ \tau_1, \tau_2, \cdots, \tau_{3g-3} \} is a disjoint union of 3g-3 ‘cuffs’.

As we can see, any simple closed curve will have an (homology) intersection number with each of the cuffs. Those numbers are non-negative integers:

Around each cuff we may assign an integer twist number, for a cuff with intersection number n and twist number z, we ‘twist’ the curve inside a little neighborhood of the cuff so that all transversal segments to the cuff will have z intersections with the curve.

Negative twists merely corresponds to twisting in the other direction:

Theorem: Every simple closed curve is uniquely defined by its intersection number and twisting number w.r.t each of the cuffs.

Conversely, if we consider multi-curves (disjoint union of finitely many simple closed curves) then any element in \mathbb{Z}^{3g-3} \times \mathbb{Z}_{\geq 0}^{3g-3} describes a unique multi-curve.

To see this we first assume that the pants decomposition comes with a canonical ‘untwisted’ curve connecting each pairs of cuffs in each pants. (i.e. there is no god given ‘0’ twist curves, hence we have to fix which ones to start with.)

In the example above our curve was homotopic to the curve ((1,2), (2,1), (1,-4)).

In other words, pants decompositions (together with the associated 0-twist arcs) give a natural coordinate chart to the set of homotopy class of (multi) curves on a surface. i.e. they are perimetrized by \mathbb{Z}^{3g-3} \times \mathbb{Z}_{\geq 0}^{3g-3}.

For the converse, we see that any triple of integers can be realized by filling the pants with a unique set of untwisted arcs:

In fact, this kind of parametrization can be generalized from integers to real numbers, in which case we have measured laminations instead of multi-curves and maximal train trucks on each pants instead of canonical untwisted arcs. i.e.

Theorem: (Thurston) The space of measured laminations \mathcal{ML}(S) on a surface S of genus g is parametrized by \mathbb{R}^{3g-3} \times \mathbb{R}_{\geq 0}^{3g-3}. Furthermore, the correspondence is a homeomorphism.

Here the intersection numbers with the cuffs are wrights of the branches of the train track, hence it can be any non-negative real number. The twisting number is now defined on a continuous family of arcs, hence can be any real number, as shown below:

As we can see, just as in the case of multi-curves, any triple of real numbers assigned to the cuffs can be realized as the weights of branches of a train track on the pants.