Indra's pearls : the vision of Felix Klein / David Mumford, Caroline Series and David Wright ; With cartoons by Larry Gonick

Felix Klein, one of the great nineteenth-century geometers, rediscovered in mathematics an idea from Buddhist mythology: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbour, so that the whole Universe was mirrored in each pearl. Klein studied infinitely repeated reflections and was led to forms with multiple co-existing symmetries. For a century these ideas barely existed outside the imagination of mathematicians. However in the 1980s the authors embarked on the first computer exploration of Klein's vision, and in doing so found many further extraordinary images. Join the authors on the path from basic mathematical ideas to the simple algorithms that create the delicate fractal filigrees, most of which have never appeared in print before. Beginners can follow the step-by-step instructions for writing programs that generate the images. Others can see how the images relate to ideas at the forefront of research. (Publisher's description).

Index

Chapter 1. The language of symmetry - an introduction to the mathematical concept of symmetry and its relation to geometric groups. Chapter 2. A delightful fiction – an introduction to complex numbers and mappings of the complex plane and the Riemann sphere. Chapter 3. Double spirals and Möbius maps – Möbius transformations and their classification. Chapter 4. The Schottky dance – pairs of Möbius maps which generate Schottky groups; plotting their limit sets using breadth-first searches. Chapter 5. Fractal dust and infinite words – Schottky limit sets regarded as fractals; computer generation of these fractals using depth-first searches and iterated function systems. Chapter 6. Indra's necklace - the continuous limit sets generated when pairs of generating circles touch. Chapter 7. The glowing gasket – the Schottky group whose limit set is the Apollonian gasket; links to the modular group. Chapter 8. Playing with parameters – parameterising Schottky groups with parabolic commutator using two complex parameters; using these parameters to explore the Teichmüller space of Schottky groups. Chapter 9. Accidents will happen - introducing Maskit's slice, parameterised by a single complex parameter; exploring the boundary between discrete and non-discrete groups. Chapter 10. Between the cracks – further exploration of the Maskit boundary between discrete and non-discrete groups in another slice of parameter space; indentification and exploration of degenerate groups. Chapter 11. Crossing boundaries – ideas for further exploration, such as adding a third generator. Chapter 12. Epilogue - concluding overview of non-Euclidean geometry and Teichmüller theory.