Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 The language of symmetry
- 2 A delightful fiction
- 3 Double spirals and Möbius maps
- 4 The Schottky dance pages 96 to 107
- 4 The Schottky dance pages 107 to 120
- 5 Fractal dust and infinite words
- 6 Indra's necklace
- 7 The glowing gasket
- 8 Playing with parameters pages 224 to 244
- 8 Playing with parameters pages 245 to 267
- 9 Accidents will happen pages 268 to 291
- 9 Accidents will happen pages 291 to 296
- 9 Accidents will happen pages 296 to 309
- 10 Between the cracks pages 310 to 320
- 10 Between the cracks pages 320 to 330
- 10 Between the cracks pages 331 to 340
- 10 Between the cracks pages 340 to 345
- 10 Between the cracks pages 345 to 352
- 11 Crossing boundaries pages 353 to 365
- 11 Crossing boundaries 365 to 372
- 12 Epilogue
- Index
- Road map
Introduction
Published online by Cambridge University Press: 05 January 2014
- Frontmatter
- Contents
- Preface
- Introduction
- 1 The language of symmetry
- 2 A delightful fiction
- 3 Double spirals and Möbius maps
- 4 The Schottky dance pages 96 to 107
- 4 The Schottky dance pages 107 to 120
- 5 Fractal dust and infinite words
- 6 Indra's necklace
- 7 The glowing gasket
- 8 Playing with parameters pages 224 to 244
- 8 Playing with parameters pages 245 to 267
- 9 Accidents will happen pages 268 to 291
- 9 Accidents will happen pages 291 to 296
- 9 Accidents will happen pages 296 to 309
- 10 Between the cracks pages 310 to 320
- 10 Between the cracks pages 320 to 330
- 10 Between the cracks pages 331 to 340
- 10 Between the cracks pages 340 to 345
- 10 Between the cracks pages 345 to 352
- 11 Crossing boundaries pages 353 to 365
- 11 Crossing boundaries 365 to 372
- 12 Epilogue
- Index
- Road map
Summary
I have discovered things so wonderful that I was astounded … Out of nothing I have created a strange new world.
János BolyaiWith these words the young Hungarian mathematical prodigy János Bolyai, reputedly the best swordsman and dancer in the Austrian Imperial Army, wrote home about his discovery of non-Euclidean geometry in 1823. Bolyai's discovery indeed marked a turning point in history, and as the century progressed mathematics finally freed itself from the lingering sense that it must describe only the patterns in the ‘real’ world. Some of the doors which these discoveries flung open led directly to new worlds whose full exploration has only become possible with the advent of high speed computing in the last twenty years.
Paralleling the industrial revolution, mathematics grew explosively in the nineteenth century. As yet, there was no real separation between pure and applied mathematics. One of the main themes was the discovery and exploration of the many special functions (sines, cosines, Bessel functions and so on) with which one could describe physical phenomena like waves, heat and electricity. Not only were these functions useful, but viewed as more abstract entities they took on a life of their own, displaying patterns whose study intrigued many people. Much of this had to do with understanding what happened when ordinary ‘real’ numbers were replaced by ‘complex’ ones, to be described in Chapter 2.
- Type
- Chapter
- Information
- Indra's PearlsThe Vision of Felix Klein, pp. xv - xxPublisher: Cambridge University PressPrint publication year: 2002