What can we do with Cayley transform? We can produce interesting pictures. Here are two such pictures:

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In fact in both pictures we have the same pattern of dots, but they are organized differently by coloring.

How are these images produced? They are produced using **Gaussian integers** and **Cayley transform**.

Cayley transform we know from the previous post Cayley transform for Easter. It is the same as in Wikipedia , where it is defined as

\begin{equation}z’=f(z)=\frac{z-i}{z+i}.\end{equation}

It maps complex upper half-plane $\mathbb{H}$, the set of all complex numbers with positive imaginary part, onto the interior $D$ of the unit disk. The real axis is mapped onto the unit circle, minus the point $z’=1.$ The inverse Cayley transform maps $1$ to infinity.

**Gaussian integers** are also explained in Wikipedia: \url{}

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. ….

We want our Gaussian integers to be in $\mathbb{H}$, or on the real line, so we take the integer defining the imaginary part to be nonnegative. To produce images above I took Gaussian integers of the form $Z=m+in$ with m varying from -100 to 100, and $n$ varying from 0 to 100. To each such $z$ I apply the Cayley transform and plot the point $f(z).$

At the end I rotate the images 90 degrees clockwise, so that the neighborhood of $z’=1$ is at the bottom. It looks for me more interesting this way.

The colors of the points are constant, either along increasing $m$ or along the increasing $n.$

Notice that the line with the same color in the first picture look like hyperbolic straight lines – they are circle segments perpendicular to the boundary.

We will need to understand why is it so?

Thank you.