## Some pictures in the hyperbolic 3-space

The following are some pictures created as part of my semester project in the 2017-18 Fall semester at EPFL. The project was under the guidance of Prof. Maryna Viazoska. The project involved building the prerequisites and understanding this paper of Don Zagier.  Below, I will give some very brief details of the pictures and the project. The full write-up that I wrote for this project can be found here.

The hyperbolic 3-space is a Riemannian manifold. As a set, the Hyperbolic 3-space ${\mathbb{H}^3}$ can be visualized as ${\mathbb{R}^2 \times \mathbb{R}_{+}}$, given the metric tensor of the Riemannian manifold given by

$\displaystyle ds^2 = (dx^2+dy^2+dt^2)/t^2. \ \ \ \ \ (1)$

The following picture is of the set given by

$\displaystyle \{(x,y,t) \in \mathbb{R}^2\times \mathbb{R}_+\ | \ x^2+y^2+t^2 \ge 1, x\le \frac{1}{2} , y \le \frac{1}{2}, x+y \ge 0 \}. \ \ \ \ \ (2)$

This set is the fundamental region of the action of the group ${PSL(2,\mathbb{Z}[i])}$ as a discrete subgroup of ${PSL(2,\mathbb{C})}$, the group of isometries on the Hyperbolic 3-space, visualized as the upper half-space of ${\mathbb{R}^3}$ in the above picture.

Measuring volumes in Hyperbolic 3-space (volume measured according to the metric tensor) is done by utilizing the concept of ideal tetrahedra. A tetrahedron is said to be an ideal tetrahedron if it has at least one ideal vertex in the upper-half space model, and one vertex at ${\infty}$, and the projection of the tetrahedron from ${\infty}$ is a right-angled triangle on the ${xy}$ plane. And ideal vertex in the upper-half space model is a vertex which is of the form of two vertical planes (spheres containing infinity), whose axis of intersection is intersecting with the unit hemispherical geodesic perpendicularly (and therefore, at the apex point of the hemisphere.

The volume of such a tetrahedron is given by the following formula in the upcoming proposition, and an illustrative picture right after.

Proposition 1 Suppose ${T_{\alpha, \gamma}}$ is the ideal tetrahedra that has a dihedral angle (the angle between the two vertical planes) as ${\gamma}$ and has an acute angle (that is, the angle between the geodesic plane opposite to ${\gamma}$ and the unit geodesic hemisphere) given by ${\alpha}$. Then the following is true.

$\displaystyle \text{Vol}(T_{\alpha,\gamma}) = \frac{1}{4}(\mathcal{L}(\alpha+\gamma)+\mathcal{L}(\alpha-\gamma)+2\mathcal{L}(\frac{\pi}{2}-\alpha)). \ \ \ \ \ (3)$

Here ${\mathcal{L}}$ is the Lobachevski function, described by the equation

$\displaystyle \mathcal{L}(\theta)= - \int_{0}^{\theta} \log|2\sin(u)| du. \ \ \ \ \ (4)$

The proof of this proposition can be found in the write-up. Here is the picture of a sample ideal tetrahedron.

For more complicated tetrahedrons, one has to decompose it by writing it as a sum and difference of ideal tetrahedrons. Any tetrahedron can be seen as the difference of two tetrahedrons with one vertex at infinity (after applying suitable isometries), both of which can be decomposed into ideal tetrahedra as shown below.

I will very crudely mention the purpose of seeing these things in a write-up about the special values of Dedekind-zeta functions.

The project that I had done entails Zagier’s generalized expression for a particular special value of the Dedekind zeta function. The generalized expression was obtained by noting the appearance of the Dedekind zeta function in the expressions of the volumes of arithmetic hyperbolic 3-manifolds, and after some careful manipulation of these volumes on arrives at the following theorem.

Theorem 2 (Zagier, 1986) Let ${K}$ be an arbitrary number field. Suppose ${A(x)}$ is a real valued function given by

$\displaystyle A(x)=\int_0^x \frac{1}{1+t^2}\log{\left( \frac{4}{1+t^2} \right)}dt. \ \ \ \ \ (5)$

Then we have that the value of ${\zeta_K(2)}$ is given by an expression as below.

$\displaystyle \zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{D}}\sum_\nu c_\nu A(x_{\nu,1})A(x_{\nu,2})\dots A(x_{\nu,n}). \ \ \ \ \ (6)$

where the sum above is finite, ${D}$ is the discriminant of the number field, ${r}$ is the number of real places and ${s}$ is the number of complex places associated to the number field ${K}$. Moreover, the ${c_\nu}$ are rational and ${x_{\mu,j}}$ are real algebraic numbers.

Here is some motivation for studying this theorem. In 1734, Euler resolved the famous Basel problem which asked the following question:

What does the sum of the following infinite series converge to?

$\displaystyle \frac{1}{1^2}+ \frac{1}{2^2}+ \frac{1}{3^2}+ \frac{1}{4^2} \dots \ \ \ \ \ (7)$

The answer was found to be ${\frac{\pi^2}{6}}$ by Euler. This has the following consequence in modern terminology. We define the Riemann-zeta function for ${1 to be the function

$\displaystyle \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}. \ \ \ \ \ (8)$

So rephrasing the Basel’s problem simply amounts to saying that ${\zeta(2)=\frac{\pi^2}{6}}$. We will consider the generalization of the identity of Equation (8) in the case of the Dedekind zeta function.

The Dedekind zeta ${\zeta_K}$ associated to the number field ${K}$ is defined as the following for a number field ${K}$ and for ${1.

$\displaystyle \zeta_{K}(s)=\sum_{ \mathcal{I} \text{ is a non-zero integer ideal of } K} N(\mathcal{I})^{-s}. \ \ \ \ \ (9)$

The Dedekind zeta function is an important function that has important connections in number theory. Putting ${s=2}$ and ${K}$ as the simplest number field ${\mathbb{Q}}$, the above equation turns into Equation (7). Zagier’s theorem is a very nice generalization of the above identity of Euler’s.

In the report, I have spent a considerable length justifying the appearance of the Dedekind zeta function in the volume formula of hyperbolic 3-manifolds.

I believe that this project was by far the most intense reading project I had ever taken.

## Some holomorphic functions

Some time ago, I saw this beautiful video shared by Prof. Terrence Tao on Google plus. The video is from the youtube channel 3Blue1Brown, where you can find some very good mathematical videos.

A much appreciable component of the particular video was the visualization done to illustrate holomorphic functions. Here’s a clip from the video:

Looking at the said video, I was inspired to make my own holomorphic function visualizer. Here are some illustrations made from my program. In the images below are visualizations of the complex plane, the blue grid is the image of the gray grid. The vertical mark on the x-axis is (1+0i).

$f(z) = e^z$

$f(z) = z^2$

$f(z) = iz$

$f(z) = \frac{1}{z}$

$f(z) = cos(z)$

$f(z) = cos(z)$                     (Again!)

An interesting thing to note here is that the perpendicularity in the lines of the grid is maintained in the image. This is a trivial consequence of the Cauchy-Riemann equations for a holomorphic function. (This fact was also mentioned in the video).

I even tried to incorporate the logarithm defined for arguments in the region $-\pi <\theta <\pi$. Here’s what it looks like:

Of course, it behaves very badly when I try to go outside my domain.

The logarithm inspires us to define our coordinates on the associated Reimann surface (or, the covering space of complex plane with the zero removed. Picture is below), instead of the 2D coordinates used in the program. The determination of which “floor” of the surface the vertex is on can be done by recording the mouse movement history. This idea could perhaps occupy me for some time in the future.

A plot of the multi-valued imaginary part of the complex logarithm function (from Wikipedia)

Another interesting discussion is that of the animation of the zeta-function in this video. In the video, the animator makes a continuous change of the grid lines to the function values. To be technical, this sort of continuous change is a homotopy of the identity function to the given function. The animator uses a simple linear homotopy, but it doesn’t look very natural.

Here’s a discussion on StackExchange about how to do the above homotopy in such a manner that the intermediate functions are holomorphic. My initial goal with the program was to try to animate that (you can spot the reminiscent code on GitHub). It is a future goal, for now.

If you have any interesting comments, or if would like to see your favorite holomorphic function animated, or if you want a better explanation of anything said above, please comment below.