Flat is Beautiful

Every flat 2-dimensional torus can be obtained by identifying opposite edges of a parallelogram.

Flatmarked 01

Each such torus has an involution that fixes four points, marked in four colors above. We can visualize the quotient as a tetrahedron with 180 degree angles at every corner by taking the triangle consisting of the lower left half of the parallelogram, and folding it together.


So the space of all tori is nothing but the space of tetrahedra. Each such tetrahedron determines a unique point on the thrice punctured sphere. This can be seen by constructing the elliptic function on the torus determined by sending red to infinity, yellow to 0, and green to 1. The unique point is then the value of blue, and is called the modular invariant of the torus. To go backwards, take a point in the thrice punctured sphere and compute the quotient of elliptic integrals (using independent cycles)

Latex image 1

This complex number is the ratio of the two edges of the parallelogram that defines the torus.
This map is a Schwarz-Christoffel map: It maps the upper half plane to a circular triangle with all angles 0.
Restricting it to the upper half disk has as its image one half of such a triangle, namely

A 0 0 x

Let’s repeat all this starting again with a parallelogram, which now has been removed from the plane.

Complement 01

Identifying again opposite edges defines a translation structure on a punctured torus that corresponds to a meromorphic 1-form with a double order zero at red (because the cone angle there is 6π), and a double order pole at infinity (yellow) (because the holomorphic 1-form dz has a double order pole at infinity). For a given modular invariant, we can determine the parallelogram to use using another Schwarz-Christoffel map

Latex image 3

which maps the upper half disk to a different circular polygon:

A 1 0 x

Curiously, we see that the ratio of the edge vectors of the parallelogram can now also lie in the lower half plane or even be real, in which case the parallelogram degenerates. For instance, we can construct a torus that corresponds to the quotient -1 by slitting the complex plane from -1 to 1, and identifying the top of the slit from -1 to 0 (resp. 0 to 1) with the bottom of the slit from 0 to 1 (resp. -1 to 0).

Slit70 01

This corresponds to an ordinary torus whose parallelogram is a rhombs with angle 70.7083 degrees. Next time we will see what this torus is good for.

Squaring the Circle

Squaring the circle is easy, you just need to know what you want to do. My personal favorite method is to use elliptic functions defined on rectangular tori to map rectangles to disks, as shown below for a square. These maps don’t preserve area (which is what the Greeks had wanted), but they preserve angles.


I had some leftover architecture images from Columbus and wanted to see how they look when made circular. Here, for instance, is the AT&T building

DSC 5596

and this is a circular version:


There are three degrees of freedom one can play with (the dimension of the automorphism group of the hyperbolic plane), which means that one can squeeze parts of the image towards the boundary cirle. Here are two other versions of the same image.


Another favorite of mine is the atrium of the Cummins office building with its wonderfully intricate play with straight lines and black and white.

DSC 5562

Now we only have to find architects and builders who create buildings that have these curves in reality.



I like it when apparently simple things evolve all by themselves into complex objects. Like watching cactus seeds grow into cacti. That was a distraction, but I do like it. Below is a left over piece of mathematics that would have fit nicely into a paper I wrote with Shoichi about triply periodic minimal surfaces.


It is, evidently, quite complicated. To unravel it, here is a smaller portion of it, its seed, so to speek.


That is a minimal surface inside a prism over a 2-3-6 triangle (which has a right angle, a 30 degree angle, and a 60 degree angle).
The curves in the vertical faces of the prism are symmetry curves of the surface, and reflecting at these faces of the prism extends the surface. The two curves in the bottom and top face of the prism are not symmetry curves, but when you place two prisms on top of each other (by translation), the curves will fit. The pattern the curves make on the prism determines the surface almost completely, there is just one degree of freedom. Here is another, equally pretty, version, using a different parameter.


Another way to seed these surfaces is through conformal geometry. Below is the conformal image of a circular annulus onto a polygonal annulus bounded by two nested 2-3-6 triangles. The parameter lines are images of radii and concentric circles, respectively. This map is the main ingredient in the Weierstrass representation of all these surfaces. Simple, isn’t it?