Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace

Mlodinow reveals how geometry's first revolution began with a "little" scheme hatched by Pythagoras: the invention of a system of abstract rules that could model the universe. That modest idea was the basis of scientific civilization. But further advance was halted when the Western mind nodded off into the Dark Ages. Finally in the fourteenth century an obscure bishop in France invented the graph and heralded the next revolution: the marriage of geometry and number. Then, while intrepid mariners were sailing back and forth across the Atlantic to the New World, a fifteen-year-old genius realized that, like the earth's surface, space could be curved. Could parallel lines really meet? Could the angles of a triangle really add up to more — or less — than 180 degrees? The curved-space revolution reinvented both mathematics and physics; it also set the stage for a patent office clerk named Einstein to add time to the dimensions of space. His great geometric revolution ushered in the modern era of physics.

Today we are in the midst of a new revolution. At Caltech, Princeton, and universities around the world, scientists are recognizing that all the varied and wondrous forces of nature can be understood through geometry — a weird new geometry. It is a thrilling math of extra, twisted dimensions, in which space and time, matter and energy, are all intertwined and revealed as consequences of a deep, underlying structure of the universe.

Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.

Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space — in the living room or in some other galaxy — have been the hidden engine of the highest achievements in science and technology.

Mlodinow reveals how geometry's first revolution began with a "little" scheme hatched by Pythagoras: the invention of a system of abstract rules that could model the universe. That modest idea was the basis of scientific civilization. But further advance was halted when the Western mind nodded off into the Dark Ages. Finally in the fourteenth century an obscure bishop in France invented the graph and heralded the next revolution: the marriage of geometry and number. Then, while intrepid mariners were sailing back and forth across the Atlantic to the New World, a fifteen-year-old genius realized that, like the earth's surface, space could be curved. Could parallel lines really meet? Could the angles of a triangle really add up to more — or less — than 180 degrees? The curved-space revolution reinvented both mathematics and physics; it also set the stage for a patent office clerk named Einstein to add time to the dimensions of space. His great geometric revolution ushered in the modern era of physics.

Today we are in the midst of a new revolution. At Caltech, Princeton, and universities around the world, scientists are recognizing that all the varied and wondrous forces of nature can be understood through geometry — a weird new geometry. It is a thrilling math of extra, twisted dimensions, in which space and time, matter and energy, are all intertwined and revealed as consequences of a deep, underlying structure of the universe.

Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.

Introduction

1. The First Revolution
2. The Geometry of Taxation
3. Among the Seven Sages
4. The Secret Society
5. Euclid's Manifesto
6. A Beautiful Woman, a Library, and the End of Civilization

7. The Revolution in Place
8. The Origin of Latitude and Longitude
9. The Legacy of the Rotten Romans
10. The Discreet Charm of the Graph
11. A Soldier's Story
12. Iced by the Snow Queen

13. The Curved Space Revolution
14. The Trouble with Ptolemy
15. A Napoleonic Hero
16. The Fall of the Fifth Postulate
17. Lost in Hyperbolic Space
18. Some Insects Called the Human Race
19. A Tale of Two Aliens
20. After 2,000 Years, a Face-lift

21. Revolution at the Speed of Light
22. Relativity's Other Albert
23. The Stuff of Space
24. Probationary Technical Expert, Third Class
25. A Relatively Euclidean Approach
26. Einstein's Apple
27. From Inspiration to Perspiration
28. Blue Hair Triumphs

29. The Weird Revolution
30. Ten Things I Hate About Your Theory
31. The Necessary Uncertainty of Being
32. Clash of the Titans
33. A Message in a Kaluza-Klein Bottle
34. The Birth of Strings
35. Particles, Schmarticles!
36. The Trouble with Strings
37. The Theory Formerly Known As Strings

Epilogue
Notes
Acknowledgments
Index