Indra's Pearls (book)

Title

Contents

• 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; identification 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.

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