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Appendix A. The Spiral of Theodorus

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The Irrationals
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appendix aThe Spiral of TheodorusWe presented the Spiral of Theodorus on page 7. In this appendixwe pursue the idea a little further for those readers whose inter-est has been aroused by the construction. First, the reader maywish to be aware that an alternative name for the spiral isQuadratwurzelschnecke, so coined in 1980 by the late Austrianmathematician Edmund Hlawka, who studied a number of itsproperties.We reproduce the original diagram below and seek a formula forthe coordinates of the non-right-angled vertices of the triangles,for which the use of complex numbers is convenient.We begin with the vertexz1=1+i and generatezn+1fromznby complex addition:zn+1=zn+the appropriate complex number ofmodulus 1 at right angles tozn.We recall that multiplying a complex number by i rotates it 90◦anticlockwise and that dividing a complex number by its modu-lus ensures that the result has modulus 1. Thus we generate the272

appendix aThe Spiral of TheodorusWe presented the Spiral of Theodorus on page 7. In this appendixwe pursue the idea a little further for those readers whose inter-est has been aroused by the construction. First, the reader maywish to be aware that an alternative name for the spiral isQuadratwurzelschnecke, so coined in 1980 by the late Austrianmathematician Edmund Hlawka, who studied a number of itsproperties.We reproduce the original diagram below and seek a formula forthe coordinates of the non-right-angled vertices of the triangles,for which the use of complex numbers is convenient.We begin with the vertexz1=1+i and generatezn+1fromznby complex addition:zn+1=zn+the appropriate complex number ofmodulus 1 at right angles tozn.We recall that multiplying a complex number by i rotates it 90◦anticlockwise and that dividing a complex number by its modu-lus ensures that the result has modulus 1. Thus we generate the272

Chapters in this book

  1. Frontmatter i
  2. Contents vii
  3. Acknowledgments ix
  4. Introduction 1
  5. Chapter one. Greek Beginnings 9
  6. Chapter Two. The Route to Germany 52
  7. Chapter Three. Two New Irrationals 92
  8. Chapter Four. Irrationals, Old and New 109
  9. Chapter Five. A Very Special Irrational 137
  10. Chapter Six. From the Rational to the Transcendental 154
  11. Chapter Seven. Transcendentals 182
  12. Chapter Eight. Continued Fractions Revisited 211
  13. Chapter Nine. The Question and Problem of Randomness 225
  14. Chapter Ten. One Question, Three Answers 235
  15. Chapter Eleven. Does Irrationality Matter? 252
  16. Appendix A. The Spiral of Theodorus 272
  17. Appendix B. Rational Parameterizations of the Circle 278
  18. Appendix C. Two Properties of Continued Fractions 281
  19. Appendix D. Finding the Tomb of Roger Apéry 286
  20. Appendix E. Equivalence Relations 289
  21. Appendix F. The Mean Value Theorem 294
  22. Index 295
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https://doi.org/10.1515/9781400841707.272

Chapters in this book

  1. Frontmatter i
  2. Contents vii
  3. Acknowledgments ix
  4. Introduction 1
  5. Chapter one. Greek Beginnings 9
  6. Chapter Two. The Route to Germany 52
  7. Chapter Three. Two New Irrationals 92
  8. Chapter Four. Irrationals, Old and New 109
  9. Chapter Five. A Very Special Irrational 137
  10. Chapter Six. From the Rational to the Transcendental 154
  11. Chapter Seven. Transcendentals 182
  12. Chapter Eight. Continued Fractions Revisited 211
  13. Chapter Nine. The Question and Problem of Randomness 225
  14. Chapter Ten. One Question, Three Answers 235
  15. Chapter Eleven. Does Irrationality Matter? 252
  16. Appendix A. The Spiral of Theodorus 272
  17. Appendix B. Rational Parameterizations of the Circle 278
  18. Appendix C. Two Properties of Continued Fractions 281
  19. Appendix D. Finding the Tomb of Roger Apéry 286
  20. Appendix E. Equivalence Relations 289
  21. Appendix F. The Mean Value Theorem 294
  22. Index 295
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