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
10 - Between the cracks pages 345 to 352
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
Slicing it to ribbons
In the mathematical world, one occasionally encounters some rare individuals who seem to have the ability to hold an image in their minds of a four or higher-dimensional object. Even the ability to visualise three-dimensional geometry is a reasonably uncommon gift, and most of us have to get by with two (and sometimes much less) dimensional images in our heads.
The space of quasifuchsian once-punctured torus groups is described by two complex or four real parameters (the traces), which means it is one of these high-dimensional objects. Our approach to studying it has been to look at two-dimensional ‘slices’, meaning that we specify one of the complex trace parameters and then plot those values of the remaining parameter which correspond to quasifuchsian groups (or single cusps, in the case of the Maskit slice).
The samples we have given, Maskit's slice and the trace 3 slice, offer an impression of a object with a somewhat pointy boundary, which is however not terribly complicated otherwise. In recent years, as more detailed plots have emerged, this simple picture has begun to change. Exactly how the slices all fit together is a puzzle of very current interest.
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- Information
- Indra's PearlsThe Vision of Felix Klein, pp. 345 - 352Publisher: Cambridge University PressPrint publication year: 2002