The author’s first motivation for this book is to chart a middle path in published works about fractals: between “colorful pictures and amazing examples” that lack mathematical depth, and professionals’ articles that are inaccessible to beginners. The conscious choice is made to focus on only two examples: the Sierpinski gasket and the Apollonian gasket. This allows for an in-depth study of two noteworthy examples of fractals.
What constitutes a “beginner” may be open to some debate; this book is deep into metric spaces and Cauchy sequences within the first 10 pages. Of course, that’s the point: making the case that fractal geometry is serious mathematics and not just a way to generate nice pictures. Among the mathematics details, a second goal emerges: to introduce young mathematicians to the kinds of questions that are the focus of mathematical research and to do so in an accessible environment. Over the course of the book, these questions about fractals are nicely connected to the bigger mathematical picture, and the two selected fractals prove a fine springboard for questions both about them and about related areas of mathematics; relations that are not readily apparent to the aficionado of colorful pictures.
Some of us in the mathematics community had our attention first drawn to fractals by these colorful pictures, perhaps by questions from friends or family who saw those pictures. In A Tale of Two Fractals, we have an excellent treatment of two important examples from a mathematically intense perspective, one which has great promise to extend our understanding.
Mark Bollman (firstname.lastname@example.org) is professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.
Part 1. The Sierpiński Gasket
Definition and General Properties
The Laplace Operator on the Sierpiński Gasket
Harmonic Functions on the Sierpiński Gasket
Part 2. The Apollonian Gasket
Circles and Disks on Spheres
Definition of the Apollonian Gasket
Arithmetic Properties of Apollonian Gaskets
Geometric and Group-Theoretic Approach
Many-Dimensional Apollonian Gaskets