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

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Summary: from Dallas Morning News
Review: "Euclid's work is a work [is] a work of beauty whose impact rivaled that of the bible, whose ideas were as radical as those of Marx and Engels. For with his book, Elements Euclid opened a window thgrough which the nature of our universe has been revealed." Strong words, but Mlodinow backs them up with this surprisingly exciting history of how mathematicians and physicists discovered geometric space beyond Euclid's three dimensions. Each advance in mathematical geometry has been followed by unexpected discoveries proving that the strange mathematics actually describe measurable physical properties. Mlodinow, a physicist and former faculty member of the California Institute of Technology, has also written TV screenplays for Star Trek: the Next Generation and other shows. He has a good sense of popular science writing, and he personalizes geometric abstractions by endowing them with personalities of his adolescent sons Alexei and Nicolai. Euclid, Descartes, Gauss, Einstein, and Witten are among the mathematicians profiled, and each of them also emerges with a distinct personality based on the style of their writing and historical anecdotes. This engaging history does an excellent job of explaining the importance of the study of geometry without requiring the reader to be a mathematician.

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Summary: Amazing Book
Review: Anyone who thought geometry was boring or dry should prepare to be amazed. Despite its worthy cover this book is exactly what its title says - a story - and the plot of this story involves life, death and revolutions of understanding and belief, and stars the some of the most famous names in history.

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?

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Summary: Brilliantly written
Review: Euclid's Window is an astounding book. It takes you on a ride through History, where you explore the origins of Geometry & Mathematics. From the early Babylonians to the Egyptians who used Geometry & Mathematics, but didn't ask the deeper questions which the Greeks did. With the Greeks came Thales, Pythagoras, and Euclid (Not necessarily in that order) who changed our view of the world by developing Geometry as we know it today in High School books. Than came the Dark Ages, and Europe plummeted into more than 1000 years of intellectual silence.

The book than talks about the revolutions led by Galileo, Descartes, Gauss, and Riemann. Finally the Author describes the later developments in physics. The revolution that Einstein made with the Special Theory or Relativity and the General Theory of Relativity. From then on we knew that mass curves space creating gravity, that nothing can travel at the speed of light, and that time is a privet matter rather than universal. Than the Revolution of Quantum Physics which, was developed in particular by Heisenberg and Schrodinger. Leonard Mlodinow explains the conflicts that arise when quantum physics and the General Theory of Relativity are combined, they fail. Quantum Mechanics works perfectly on the small scale, General Relativity works fine on the large scale, yet there is no way physicist and mathematicians could combine the two.

And then came the birth of the String Theory, rather five different String theories that turn out to be approximations to the much larger M-Theory, the theory that would be able to describe everything in the Universe, from subatomic particles to distant galaxies in the Universe. There is only one problem nobody knows what it looks like, and mathematicians and physicist can only calculate approximations of the theory. Leonard Mlodinow takes the reader on a fascinating ride through the history of Mathematics and Physics, the book is enlightening, even to me who constantly tries to keep up with new developments in Physics.

A must read for anybody interested in Mathematics & Science, or just plain old History, this book is essential.