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
4 - The Schottky dance pages 107 to 120
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
Tiles and pretzels
Thus far, our symmetry groups have always gone hand in hand with a tiling; either the bathroom floor tiles in the picture on p. 17 in Chapter 1, or the ring-shaped tiles associated to the group generated by a single loxodromic transformation in Chapter 3. As we already hinted, there is still a set of symmetrical tiles hidden amid the circles in the Schottky array. It is a bit harder to spot them, because their shape is a rather more exotic than those we have met thus far. Perhaps a volunteer could help us. Dr. Stickler, would you do us the favour of guiding us around?
Figure 4.7 shows a time elapsed photograph of Dr. Stickler's journey. Starting boldly at the centre of the picture, you see him shrinking and turning as he progresses ever more deeply into the Schottky array. A new Dr. Stickler appears for each transformation in the group. The most striking feature of the picture is that the arrows exactly reflect the pattern in the word tree. Each Dr. Stickler is at a node, and if you copied over the labelling from the word tree you would find each copy labelled by the exact transformation which gets him from his central position to the given location. Of course, we only show a finite number of levels, but it is not hard to imagine the whole infinite tree of words extending outwards, the edges getting ever shorter until we eventually reach a limit point at the infinite end of the branch.
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- Indra's PearlsThe Vision of Felix Klein, pp. 107 - 120Publisher: Cambridge University PressPrint publication year: 2002