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
8 - Playing with parameters pages 245 to 267
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
Hitting the edge
Figure 8.14 shows two pictures in which one of the two traces is 2 and the other is 3. In other words, one of the two generators a and b is parabolic but the other is not. Both pictures are rather like the gasket picture in frame (vi) of Figure 8.2, but on the left only the circles Ca and CA have come together with an extra point of tangency, while on the right the tangency is between the circles Cb and CB. This may be easier to see if you compare the left frame of Figure 8.14 to Figure 8.4. See how the fixed points of a have come together pinching off the lefthand part of the picture from the right. If, on the other hand, we fix ta and send tb to 2, the upper and lower pincers come together resulting in the righthand frame of Figure 8.14.
The myriad small circles in these pictures appear for exactly the same reason as they did in the last chapter. If for example a is parabolic, then so is bAB, and so also is abAB. Thus the two elements a,bAB generate a subgroup conjugate to the modular group, which as we know means we expect to see circles in the limit set. Well, here they are!
Groups like these in which one element is parabolic are called cusp groups, because they can be explained in terms of pinching points on surfaces to cusps. Some groups, like the ones in our pictures here, have one ‘extra’ parabolic element (in this case b) and so are called single cusps.
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- Information
- Indra's PearlsThe Vision of Felix Klein, pp. 245 - 267Publisher: Cambridge University PressPrint publication year: 2002