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

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

(1)

It maps complex upper half-plane , the set of all complex numbers with positive imaginary part, onto the interior of the unit disk. The real axis is mapped onto the unit circle, minus the point The inverse Cayley transform maps 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 , 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 with m varying from -100 to 100, and varying from 0 to 100. To each such I apply the Cayley transform and plot the point

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

The colors of the points are constant, either along increasing or along the increasing

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?