
This is a 3D version of the Apollonian depth fractal, introduced here. The movie shows (x,y) plane and time goes through the section along the z axis.

How do you add velocities?
In the theory of relativity, velocities need to be added not according to the simple Galilean law:
but rather according to Poincare's formula:
Not intuitive as it is, it still may be presented in the form of a geometric diagram:
Here is an interactive version of this diagram. Have fun.
More you can read in my
preprint.
It contains also a diagram for trigonometric tangent, unlike the
published version (Am. J. Phys. Aug 2012 80 (8) p. 737.)

Next week it is my turn to tell something at our qolloquium. The above is the title of my talk. Below is the abstract. In a meantime you may want to play with this interactive applet [CLICK on the image to go to a new page]:
Abstract: Quite interestingly, a number of key ingredients of mathematical physics appear in some form in the Apollonian Window, a special case of an Apollonian disk packing. We will present some of its numbertheoretic features, analogies with the theory of relativity, and then concentrate on the spinor structure concealed in it. A sequence of theorems will explain the flow of the spinors in the gasket including the topological nontriviality of the spinor fiber bundle over the gasket.
You may also visit my page devoted to the
Apollonian Window.