Links for Proofs from Book

  1. Paul Erdos
  2. Euclid
  3. Art Gallery
  4. sperner's lemma
  5. Fixed points, Brouwer's fixed point theorem
  6. Leonhard Euler
  7. Euler's formula
  8. Pick's theorem
  9. Continuum Hypothesis
    The Continuum Hypothesis:
    Continuum Hypothesis: True, False, or Neither?
    Intuitivistic solution of Continuum Hypothesis
    Warning this doesn't look like ``real'' mathematics, especially since the url is the author's name.
    The Continuum Hypothesis:
    Some formulations of Continuum Hypothesis
    Georg Cantor
    Cantor's Biography
    Godel, Kurt
    Cohen, Paul J.
  10. Here are some links concerning irrational numbers. The first is temporarily out of order, but the caretakers have suggested it will be back up soon.
    Mathematical Constants (besides e and pi)
    Millions of Digits of a Few Favorites
    How Irrational Can an Irrational Be? (relations to bio)
  11. Euler-Lagrange
  12. Chapter 7 Galileo's "Miraculous" Geometry Problem
  13. Page with Lots of Links Relating to Polyhedra
  14. Chapter 10 Seventeen (!) Proofs of Euler's Formula
  15. Chapter 21 Pigeon-Hole Lesson + Exercises
  16. David Hilbert (biography)
  17. Hilbert's problems
  18. Hilbert's thirth problem/The Dehn-Hadwiger theorem
  19. Graph theory
  20. Graph Coloring
  21. Carsten Thomassen
  22. Margit Voigt
  23. 4-color problem
  24. Sperner's Theorem
  25. chapter 22
  26. Bernoulli Numbers
  27. Bernoulli +tml
  28. Bernoulli numbers
  29. Joseph Bertrand
  30. Bertrand's postulate
  31. One (and a Half) More Proofs of the Marriage Theorem
  32. Bernoulli
  33. Bertrand' Postulate
  34. Srinivasa Aaiyangar Ramanujan
  35. Pafnuty Lvovich Chebyshev
  36. Jacob Bernoulli
  37. Bernoulli Numbers
  38. Gustav Herglotz
  39. ch19/2
  40. Happy Ending
  41. Ramsey Numbers
    Quick overview, typical of Mathworld:
    Very nice paper on the state of what is known:
    Here's a page on an overview of cardinals. Ramsey Cardinals are barely mentioned, but I like it as a quick reference:
  42. Ramsey Numbers
  43. Prufer Codes
  44. Cayley's Formula
  45. Pafnuty Chebyshev
  46. Chebyshev polynomials etc
  47. George Polya
  48. lBibliography of Bernoulli Numbers, for Those Who Would Know More
  49. A Virtual Compendium of Bernoulli Number Information
  50. The Bernoulli Dossier: Short Bios
  51. Pierre de Fermat
  52. Sums of squares
  53. J.J. Sylvester
  54. Sylvester's line problem etc,syl.html
  55. On Frank Ramsey:
  56. Erdos quote on Ramsey numbers:

    Erdos related the following anecdote: "Aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five [that is, R(5,5)]. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack. (Graham, Ronald L. and Joel H. Spencer. Ramsey Theory. Scientific American July 1990: 112-117).

  57. On Ramsey numbers:,ramsey.html
  58. Stirling's Formula
  59. Stirling (the man)
  60. Waring's problem
  61. Variant of Waring's Problem
  62. Visualization Method for Inspiring Intuition (discusses Waring's Problem)
  63. J. E. Littlewood,,2000,00.html
  64. D. J. Kleitman,+D.+J.
  65. Paul Erdos
  66. An Imaginary Tale:
    The Story of the square root of minus one's_formula
  67. Harmonic series and gamma constant
  68. CHAPTER 17
  69. CHAPTER 18
  70. CHAPTER 29
  71. CHAPTER 8
  72. Proofs and Refutations
  73. Origami and math
    How to Make Origami Gifts Thanks to Carly Waters' Class at Amorita Charter School
    Paper Art Reproductions: Learn Origami! Thanks to Joan Ward's Art Class in Maine.
  74. An unearthed Lakatos lecture (mp3 or transcript):
  75. This doesn't really relate to P&R, but here's a paper on L's philosophy of science. It's pretty dry, but I'll mention it anyway. I have a book of some of his stuff on epistomology somewhere, but I'll bypass further philosophical refs.
  76. Anagrams of Imre Lakatos: "So I'm a talker." and "OK at realism?"
  77. The Moore Method:
  78. CH20: Buffon's needle problem (French)
  79. I like this overview: (ch 15 topis)
  80. infinite ink is a weird site. I don't yet know what to make of it, but here's their CH stuff. It's all pretty sound, but reading their site always gives me this creepy cult feeling.
  81. This paper is a response to Woodin's papers on the CH, which you already have posted.
  82. biographical, Georg Cantor:
  83. a nice little article summarizing CH and related issues. I like this line: "When a mathematician finds himself supporting two contradictory propositions, he's obviously been a department chair or a dean for too long and it's time to give up and move on."
  84. Not entirely relevant, but this site is worth mentioning (and the cont. hyp. is briefly covered). It's called "interactive real analysis," and.. well, that's pretty much what is. Hypetext notes with some widgets.
  85. Even less relevant, but funny, is that the contiuum hypothesis is related to the Navier-Stokes Equations, a Millenium Problem! Unfortunately, it's not THE contiuum hypothesis, but rather a much less interesting assumption about fluid mechanics. Silly physicists. Anyway:
  86. Chapter 24
  87. Arthur Cayley bio
  88. Jim Pitman's home page
  89. Chapters 24/17
  90. George Polya bio
  91. Polya problem-solving method
  92. Polya Essay on Math Education
  93. Chapter 17
  94. Chebyshev bio
  95. Chebyshev's Inequalities
  96. Chapter 8
  97. J.J. Sylvester, author of "The Laws of Verse"
  98. Unsolved Problems in Graph Theory (including Gallai conjecture)
  99. An American i Tale
  100. Imaginary Numbers and Spacetime
  101. All About i
  102. Chapter 29
  103. Paul Turan bio
  104. Erdos eulogizes Turan
  105. Chapter 18
  106. On Littlewood
  107. Proof of Cauchy-Schwartz Inequality,csi.html
  108. David Hilbert
  109. Chapter 20
  110. Buffon's Needle with Simulation
  111. Not Just Dropping Needles: Buffon the Revolutionary
  112. Chapter 5
  113. To Witt (Ernst Witt, that is)