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
6 - Indra's necklace
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
Beautiful is thy wristlet, decked with stars and cunningly wrought with myriad-coloured jewels.
Gitanjali, Rabindranath TagoreThe picture in Figure 6.1 shows our first truly impressive fractal construction created by the simultaneous symmetry of two Möbius maps. We made it by introducing what seems like only a small change in the set-up of the previous two chapters: we just let the four circles come together until they were tangent, forming, as we shall say, a kissing chain. In a beautiful way, this causes the limit set to coalesce from dust into a ‘necklace’. If you look carefully, you will see how the nested Schottky circles form smaller and smaller chains of tangent circles. Instead of nesting down on fractal dust, these chains shrink down onto a glowing and rather curiously crinkled loop. The nesting circles seem to condense near the tangency points of the circles, highlighted by our colour coding like brilliant blue jewels. Except for the fact that the circles are tangent, this picture is entirely similar to the one in Figure 5.1.
To see better what is going on, in Figure 6.2 we zoomed in to the small black rectangle marked in the large scale picture. Now you can see more clearly the intricate complexity with which this marvellous necklace has been fashioned. Near the bottom you get a fairly clear view of some of the inner levels of chains of kissing circles, while near the main tangency point, nested tangent circles pile up creating a wonderful jewelled effect only hinted at in Figure 6.1.
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- Chapter
- Information
- Indra's PearlsThe Vision of Felix Klein, pp. 157 - 195Publisher: Cambridge University PressPrint publication year: 2002