PREFACE
1. Quoted in Schechter (1998), 155.
INTRODUCTION
1. Quoted in Machamer (1998).
2. Juškevič and Winter (1965), 333.
CHAPTER 1. LEONHARD EULER AND HIS THREE ”GREAT” FRIENDS
1. Quoted in Dunham (1999), xiii.
2. Quoted in Youschkevitch (1971)
3. Riasanovsky (1993), 285.
4. Vucinich (1963), 69.
5. Quoted in Condorcet (1786)
6. Quoted in Eves (1969b), 48.
7. Quoted in Boyer and Merzbach (1991), 440.
8. Quoted in Cajori (1927)
9. Quoted in Calinger (1996)
10. Quoted in Cajori (1927)
11. Riasanovsky (1993), 248.
12. Quoted in Alexander (1989), 173.
13. Weil (1984)
14. Hartley (2003)
15. Hardy (1992), 70.
16. Vucinich (1963), 146–47.
17. Condorcet (1786)
18. Wells (1990)
CHAPTER 2. WHAT IS A POLYHEDRON?
1. Hemingway (1932), 122.
2. Francese and Richeson (2007)
3. Poincaré (1913), 434.
CHAPTER 3. THE FIVE PERFECT BODIES
1. McEwan (1997), 20.
2. Plato (1972), 244.
3. Waterhouse (1972)
4. Ibid.
CHAPTER 4. THE PYTHAGOREAN BROTHERHOOD AND PLATO’S ATOMIC THEORY
1. Simmons (1992), 20.
2. Burkert (1972), 109.
3. Quoted in van der Waerden (1954), 94.
4. Quoted in Euclid (1926) vol. 3, 438.
5. Quoted in van der Waerden (1954), 165.
6. Taylor (1929), 5.
7. Boyer and Merzbach (1991), 84.
8. Allan (1975)
9. Plato (2000), 46.
CHAPTER 5. EUCLID AND HIS ELEMENTS
1. Russell (1967), 37–38.
2. Quoted in Bulmer-Thomas (1976)
3. Ibid.
4. van der Waerden (1954), 173.
5. Euler (1862)
6. Cauchy (1813a)
7. Connelly (1977)
8. Quoted in Bulmer-Thomas (1967), 195.
CHAPTER 6. KEPLER’S POLYHEDRAL UNIVERSE
1. Simmons (1992), 69.
2. Koestler (1963), 262.
3. Ibid, 252.
4. Kepler (1596), English translation in Kepler (1981)
5. Kepler (1596), quoted in Gingerich (1973)
6. Kepler (1981), 107.
7. Quoted in Martens (2000), 146.
8. Kepler (1938), English translation in Kepler (1997)
9. Quoted in Emmer (1993)
CHAPTER 7. EULER’S GEM
1. Bell (1987), 16.
2. Juškevič and Winter (1965), 333.
3. Euler (1758b)
4. Legendre (1794)
5. Juškevič and Winter (1965), 333.
6. Ibid., 334.
7. Euler (1758b)
8. Ibid.
9. Euler (1758a), English translation in Euler (1758c)
10. Legendre (1794)
11. Euler (1758a), English translation in Euler (1758c)
12. Lebesgue (1924)
13. Francese and Richeson (2007); Samelson (1996)
CHAPTER 8. PLATONIC SOLIDS, GOLF BALLS, FULLERENES, AND GEODESIC DOMES
1. Bell (1945), 211.
2. Poincaré (1913), 44.
CHAPTER 9. SCOOPED BY DESCARTES?
1. Descartes (1965), 259.
2. Quoted in Bell (1937), 35.
3. Descartes (1965)
4. Kuhn (1970), 54.
5. Quoted in Federico (1982), 76.
6. Lebesgue (1924)
7. Kuhn (1970), 55.
CHAPTER 10. LEGENDRE GETS IT RIGHT
1. Albers (1994)
2. Lohne (1972)
3. Quoted in Itard (1972)
4. Girard (1629)
5. Quoted in Itard (1972)
6. Poinsot (1810)
7. Ibid.
CHAPTER 11. A STROLL THROUGH KÖNIGSBERG
1. Thoreau (1894), 419.
2. Quoted in Sachs, Stiebitz, and Wilson (1988)
3. Ibid.
4. Quoted in Hopkins and Wilson (2004)
5. Euler (1736), English translation in Biggs, Lloyd, and Wilson (1986), 3–8.
6. Ball (1892)
7. Hierholzer (1873)
8. Barabási (2002), 12.
9. Listing (1847)
10. Terquem (1849)
CHAPTER 12. CAUCHY’S FLATTINED POLYHEDRA
1. Abel (1881), 259.
2. Freudenthal (1971)
3. Ibid.
4. Simmons (1992), 186.
5. Cauchy (1813a)
6. Lhuilier (1813)
7. Ibid.
8. Hadamard (1907)
9. Listing (1861–62); Jordan (1866b)
CHAPTER 13. PLANAR GRAPHS, GEOBOARDS, AND BRUSSELS SPROUTS
1. Hankel (1884)
2. Hardy (1992), 94.
3. Pick (1899)
4. DeTemple (1989)
5. Gardner (1975a), 8.
6. Applegate, Jacobson, and Sleator (1991)
CHAPTER 14. IT’S A COLORFUL WORLD
1. Twain (1894), 42–43.
2. May (1965)
3. Graves (1889), 423.
4. Ibid.
5. Cayley (1878)
6. Quoted in Dudley (1992)
7. Gardner (1975b); Gardner (1988)
8. Baltzer (1885), quoted in Coxeter (1959)
9. Ibid.
10. Kempe (1879)
11. Quoted in Wilson (2002), 119.
12. Gardner (1995)
13. Appel and Haken (1977); Appel, Haken, and Koch (1977)
14. Hales (2005)
CHAPTER 15. NEW PROBLEMS AND NEW PROOFS
1. Quoted in Federico (1982), 71.
2. Sommerville (1958), 143–44.
3. de Jonquières (1890)
4. Speziali (1973)
5. Pont (1974), 24.
6. Lhuilier (1813)
7. Hessel (1832)
8. Poinsot (1810)
9. Cauchy (1813a)
10. Lhuilier (1813)
11. Steiner (1826)
12. von Staudt (1847), 18–23.
13. Hoppe (1879)
CHAPTER 16. RUBBER SHEETS, HOLLOW DOUGHNUTS, AND CRAZY BOTTLES
1. Listing (1847)
2. Tait (1883)
3. Lefschetz (1970)
4. In an interview in Maurer (1983)
5. Klein (1882/83)
6. Brahana (1921)
7. Clarke (2000)
8. Gardner (1990)
9. Gardner (1956)
10. Listing (1861–62)
11. Möbius (1865)
12. In the introduction of Abbott (2005), xxix.
13. Klein (1882)
CHAPTER 17. ARE THEY THE SAME, OR ARE THEY DIFFERENT?
1. Poincaré (1895)
2. Möbius (1863)
3. Radó (1925)
4. Papakyriakopoulos (1943)
5. Quoted in Freudenthal (1975)
6. Riemann (1851); Riemann (1857)
7. Möbius (1863)
8. Jordan (1866a)
9. Dyck (1888)
10. Dehn and Heegaard (1907)
11. Francis and Weeks (1999)
12. Ibid.
1. Shakespeare (2002), 82.
2. Vandermonde (1771)
3. Gauss (1877)
4. Listing (1847)
5. Seifert (1934)
6. Sequence number A002864 in Sloane (2007)
7. Crowell (1959)
8. Sequence number A002863 in Sloane (2007)
9. Kauffman (1987b); Murasugi (1987); Thistlethwaite (1987)
CHAPTER 19. COMBING THE HAIR ON A COCONUT
1. Frost (2002), 308.
2. Brouwer (1912)
3. Quoted in Dieudonné (1975)
4. Dieudonné (1975)
5. Poincaré (1881)
6. Poincaré (1885)
7. Brouwer (1912)
8. Beno Eckmann, quoted in Frei and Stammbach (1999)
9. Hopf (1925); Hopf (1926a); Hopf (1926b)
10. Morse (1929)
11. Thurston (1997)
12. Brouwer (1909)
13. Brouwer (1912)
CHAPTER 20. WHEN TOPOLOGY CONTROLS GEOMETRY
1. Shakespeare (1992), 36.
2. Pólya (1954), 57–58.
3. Hopf (1935)
4. Quoted in Federico (1982), 43.
5. Euler (1758b); Euler (1758a)
CHAPTER 21. THE TOPOLOGY OF CURVY SURFACES
1. Bell (1937), 254
2. Euler (1760)
3. See Hayes (2006) for a discussion of this story.
4. Quoted in Simmons (1992), 177.
5. Gauss (1828); English translation and commentary in Dombrowski (1979)
6. Bonnet (1848)
7. Blaschke (1921)
8. Dyck (1888)
CHAPTER 22. NAVIGATING IN n DIMENSIONS
1. Scholz (1999)
2. Brouwer (1911)
3. Cauchy (1813a)
4. Schläfli (1901)
5. Breitenberger (1999)
6. Listing (1847), Listing (1861–62)
7. Tait (1884)
8. Riemann (1851)
CHAPTER 23. HENRI POINCAKÉ AND THE ASCENDANCE OF TOPOLOGY
1. Hardy (1992), 85.
2. Quoted in Dieudonné (1975)
3. Poincaré (1895)
4. Poincaré (1899); Poincaré (1900); Poincaré (1902a); Poincaré (1902b); Poincaré (1904)
5. Dieudonné (1989), 17.
6. Heinrich Tietze (1880–1964), quoted in James (2001)
7. Poincaré (1895)
8. Poincaré (1895), quoted in Sarkaria (1999).
9. Quoted in James (1999)
10. Ibid.
EPILOGUE: THE MILLION-DOLLAR QUESTION
1. Russell (1957)
2. Poincaré (1900)
3. Poincaré (1904)
4. Ibid.
5. Taubes (1987)
6. Smale (1961)
7. Smale (1990)
8. Ibid.
9. Stallings (1962); Stallings (1960); Zeeman (1961); Zeeman (1962)
10. Freedman (1982)
11. Smale (1998)
12. Thurston (1982)
13. Hamilton (1982)
14. Perelman (2002); Perelman (2003b); Perelman (2003a)
15. Cao and Zhu (2006a); Cao and Zhu (2006b); Kleiner and Lott (2006); Morgan and Tian (2006)
16. Mackenzie (2006)
17. See Nasar and Gruber (2006) for more details.
18. Nasar and Gruber (2006)
19. Quoted in Nasar and Gruber (2006)
20. Poincaré (1913), 366.