Euclid's Window

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.

The book opens with Aristotle watching ships at sea disappearing hull first over the horizon. "On a flat earth, ships should dwindle evenly until they disappear", and so he came to the realisation that the earth must be curved. This sets the scene for Mlodinow's tale of how geometry has shaped human history - "to observe the large scale structure of our planet, Aristotle had looked through the window of geometry." The book recounts how we have continued to look through this window to understand the reality we live in, and how the window has changed along the way.

The book is arranged as a series of five tales of the "five geometric revolutions of world history". These are told as the story of their main figures - Euclid, Descartes, Gauss, Einstein and Witten - in the context of their time, place and culture. This is one of the things that makes this book stand apart from others on the history of mathematics and science. It is told as a series of personal stories, of discoveries and leaps of understanding made by human beings. And this perhaps unexpectedly human side of geometry is enhanced by Mlodinow's accessible style. He is able to bring historical situations and mathematical concepts to life with the language of the present day. For example he explains the importance of applied geometry to Egyptians: "In building a pyramid, just a degree off from true, and thousands of tons of rocks, thousands of person-years later, hundreds of feet in the air, the triangular faces of your pyramid miss, forming not an apex by a sloppy four pointed spike. The Pharaohs, worshipped as gods, with armies who cut the phalluses off enemy dead just to help them keep count, were not the kind of all-powerful deities you would want to present with a crooked pyramid."

This book also contains some of the clearest explanations of relativity and string theory that I have ever read. Placed in the context of the evolution of geometry, and told as human triumphs of discovery by Einstein and Witten and their peers, these theories offer answers to obvious questions arising from our struggle to understand our reality. They also contain some very amusing examples such as Mlodinow explaining the entropy of black holes in terms of the messiness of his son, Alexei's bedroom. "Before Hawking, black holes, thought to have no internal structure, were thought to be something like an empty room. But now it seems they are like Alexei's actual room. Had Hawking asked, I could have confirmed this: I have always told Alexei that his room was like a black hole."

This is an excellent book not just for those select few fascinated by geometry, but for anyone interested in history of science, philosophy and humanity. In fact I would recommend it to anyone who enjoys a good story. Who would have thought that the story of geometry would include tales of life, death, sex and taxes?

Starting with the mathematicians and geometers of antiquity, Mlodinow traces the progress of rational thought - and irrational numbers - from before Euclid's elucidation of the Elements of geometry to the possibilities which still wait for us to reveal them - from "A point is that which has no part" straight up to the equally puzzling notion that space and time may only be shadowy hints of some more fully flowering, if abstract, function of mathematics on another plane of reality. Sound like science fiction? Rest assured that Mlodinow has both feet planted square on terra firma. The paradoxes and upsets of his discipline are not lost on the author - nor, indeed, are the ironies and jokes of history (say what you like about death, but it was the decidedly un-mystical necessity of taxation which launched geometry as a scholarly pursuit in ancient Egypt) - but the author reminds his reader at various points of the dangers of assuming too readily that any given idea is worthless, too far-out, or obviously and intuitively wrong. Intuition, as it turns out, resists and rebels against much of what has become higher learning in the fields of mathematics and physics.

Mlodinow's dedication to the subject matter at hand matches in beautiful, if heartbreaking, counterpoint to the obscurity in which many of the scholars he discusses labored. Drawing not only on the work of famous theoreticians like Einstein and Hawking, but also on essays and ideas buried in forgotten papers and musty appendices, the author gives full credit wherever it may be due. In the process, whether by design or accident, Mlodinow imparts an even more valuable lesson: the ease with which scientific knowledge can be lost, sometimes for millennia. If Artistotle knew, nearly 2,500 years ago, that the planet must be round, why do we still hear that Columbus' sailors were terrified of sailing off the edge of a flat Earth? (This story in itself is almost certainly apocryphal.) If primitive versions of the Theory of Evolution were kicking around in ancient Greece, how is it we still face voids of serious scientific credibility in modern-day Kansas? Regrettably, superstition, fear, politics, and the manipulation of knowledge - who gets it and who pays the price for seeking too much of it - is also part of the history of geometry, as it is part of the history of science in general.

Your reviewer himself studied a fair amount of the history of mathematics and physics in the Western World (starting, in fact, with Euclid, and progressing then through Ptolemy, Apollonius, Descartes, Newton, et al, right up through Einstein and Minkowski) and found certain parts of the curriculum cheerless, if not downright appalling. What a relief and a joy, then, to find Euclid's Window not only concise and readily understandable, but effervescent as well. Author Mlodinow clearly enjoys the subject matter and - more importantly - enjoys imparting it to others. As a writer and a teacher, Mlodinow demonstrates that he is gifted and enthusiastic.

It's hard to want to be critical of a book like this, which is both charming and brave in the face of apathy, even hostility, toward mathematics and scientific inquiry (a situation far from unique to our times). Indeed, there is little to be critical about here, for the book is nearly perfect in its balance of detail and simplification. If anything, though, the simplification of the material may be a little too rigorously carried out, and the focus on geometry a bit too narrow. Though quantum physics is not the focus of the book, it has become relevant to contemporary geometric notions, and while Mlodinow does incorporate plenty on the field and its proponents, the origins of quantum physics are glossed over a bit too much. A broad sketch of the theories underlying quantum physics would not have been out of place, but there is not so much as a mention of black-body radiation, the effect which first put physicists onto the notion of quantum energy states. As if in compensation, however, Mlodinow's explanation of relativity strikes a perfect balance and his exposition on string theory is wonderfully clear. Don't be scared off if these sound like high-falutin subjects impossibly out of reach: Mlodinow does the invaluable service of grasping a higher limb of the Tree of Knowledge and bending it down until the layman can get a hold on, and enjoy, its fruits.

There are some very good points about this book, which include the presentation of the original developments of geometry very clearly and in a very highly understandable manner. The book is also full of little trivia and delves cursorily into some of the aspects of the personalities behind the mathematics. But it flounders as it reaches string theory and becomes very abstract. Perhaps the author could have spent lesser time in the initial parts by cutting out some of the analogies he would draw with the charactors in the example bearing his childrens names.

Having said that, this is a good book and is worth a buy for the Maths lover. In terms of analogies, if you liked Eli Maors "e The story of a number", Mlodinov will fit in well in his ease of development of a complex history.