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### Summary

### Summary

The fascinating story of a problem that perplexed mathematicians for nearly 400 years

In 1611, Johannes Kepler proposed that the best way to pack spheres as densely as possible was to pile them up in the same way that grocers stack oranges or tomatoes. This proposition, known as Kepler's Conjecture, seemed obvious to everyone except mathematicians, who seldom take anyone's word for anything. In the tradition of Fermat's Enigma, George Szpiro shows how the problem engaged and stymied many men of genius over the centuries--Sir Walter Raleigh, astronomer Tycho Brahe, Sir Isaac Newton, mathematicians C. F. Gauss and David Hilbert, and R. Buckminster Fuller, to name a few--until Thomas Hales of the University of Michigan submitted what seems to be a definitive proof in 1998.

George G. Szpiro (Jerusalem, Israel) is a mathematician turned journalist. He is currently the Israel correspondent for the Swiss daily Neue Zurcher Zeitung.

### Author Notes

GEORGE G. SZPIRO, Ph.D., is a mathematician turned journalist. He is currently the Israel correspondent for the Swiss daily Neue Zurcher Zeitung.

### Reviews 1

### Choice Review

Mathematics often is viewed as an edifice of problems that need to be solved, many of which capture the public's attention. When one of these "gems" is solved, attention turns to another problem. For example, once both the four-color conjecture and Fermat's last problem were solved recently, the focus shifted to new "classics," such as Sir Walter Raleigh's question: what is the most efficient way to stack cannonballs on a ship? In 1611, astronomer Johannes Kepler claimed the solution was to stack them the natural way, which is how grocers stack spherical fruit. Unfortunately, though most mathematicians felt that Kepler's conjecture was correct, its justification proved to be quite stubborn. This book, using a delightful and engaging style, describes the 400-year effort to either justify or disprove Kepler's conjecture, with young mathematician Thomas Hales producing a "solution" in 1998. Though Kepler's conjecture and its resolution have not received as much press or attention as Wiles's solution to Fermat's problem, the story is equally interesting. Using an engaging historical context, Szpiro makes the mathematics understandable, relegating the more difficult material to appendixes. Helpful reference list. ^BSumming Up: Highly recommended. General readers; lower-division undergraduates through faculty. J. Johnson Western Washington University

### Table of Contents

Preface | p. v |

1 Cannonballs and Melons | p. 1 |

2 The Puzzle of the Dozen Spheres | p. 10 |

3 Fire Hydrants and Soccer Players | p. 33 |

4 Thue's Two Attempts and Fejes-Toth's Achievement | p. 49 |

5 Twelve's Company, Thirteen's a Crowd | p. 72 |

6 Nets and Knots | p. 82 |

7 Twisted Boxes | p. 99 |

8 No Dancing at This Congress | p. 112 |

9 The Race for the Upper Bound | p. 124 |

10 Right Angles for Round Spaces | p. 140 |

11 Wobbly Balls and Hybrid Stars | p. 156 |

12 Simplex, Cplex, and Symbolic Mathematics | p. 181 |

13 But Is It Really a Proof? | p. 201 |

14 Beehives Again | p. 215 |

15 This Is Not an Epilogue | p. 229 |

Mathematical Appendixes | |

Chapter 1 p. 234 | |

Chapter 2 p. 238 | |

Chapter 3 p. 239 | |

Chapter 4 p. 243 | |

Chapter 5 p. 247 | |

Chapter 6 p. 249 | |

Chapter 7 p. 254 | |

Chapter 9 p. 258 | |

Chapter 11 p. 263 | |

Chapter 13 p. 264 | |

Chapter 15 p. 279 | |

Bibliography | p. 281 |

Index | p. 287 |