The Apollonian Spider V - The Magic
This is the final part of this little series on the Apollonian gasket or Kleinian fractals in more general terms. In this part, attention is paid to the mathematical concepts that lie underneath.
After all it is a nice application of group theory. Unlike many other applications, this one relies on a discrete sub group of SL2(C) with infinitely many elements. It is a so-called freely generated group, whose limit set can be identified with fractals in general, and with the Apollonian gasket in a particular case.
content:
0:00 Introduction
1:10 The dihedral group - A group tree construction
4:55 The free group of the Apollonian gasket
6:36 The fundamental domain
8:30 The simplified disc representation and the group tree
10:05 Fixed points revisited
10:55 The single fixed points of a and b
11:51 The single fixed point condition
12:45 The tree extension
16:15 Fixed points replacements
17:35 Toy implementation
20:30 Outro