Summer Reading B2G1 Free
 
 

Special Offers see all

Enter to WIN a $100 Credit

Subscribe to PowellsBooks.news
for a chance to win.
Privacy Policy

Visit our stores


    Recently Viewed clear list


    Original Essays | July 14, 2015

    Joshua Mohr: IMG Your Imagination, Your Fingerprint



    When I was in grad school, a teacher told our workshop that if a published novel is 300 pages, the writer had to generate 1,200 along the way. I... Continue »
    1. $17.50 Sale Hardcover add to wish list

      All This Life

      Joshua Mohr 9781593766030

    spacer
Qualifying orders ship free.
$8.95
List price: $16.00
Used Trade Paper
Ships in 1 to 3 days
Add to Wishlist
Qty Store Section
1 Beaverton Mathematics- Geometry and Trigonometry

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

by

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

 

 

Excerpt

Introduction

Twenty-four centuries ago, a Greek man stood at the sea's edge watching ships disappear in the distance. Aristotle must have passed much time there, quietly observing many vessels, for eventually he was struck by a peculiar thought. All ships seemed to vanish hull first, then masts and sails. He wondered, how could that be? On a flat earth, ships should dwindle evenly until they disappear as a tiny featureless dot. That the masts and sails vanish first, Aristotle saw in a flash of genius, is a sign that the earth is curved. To observe the large-scale structure of our planet, Aristotle had looked through the window of geometry.

Today we explore space as millennia ago we explored the earth. A few people have traveled to the moon. Unmanned ships have ventured to the outer reaches of the solar system. It is feasible that within this millennium we will reach the nearest star — a journey of about fifty years at the probably-some-day-attainable speed of one-tenth the speed of light. But measured even in multiples of the distance to Alpha Centauri, the outer reaches of the universe are several billion measuring sticks away. It is unlikely that we will ever be able to watch a vessel approach the horizon of space as Aristotle did on earth. Yet we have discerned much about the nature and structure of the universe as Aristotle did, by observing, employing logic, and staring blankly into space an awful lot. Over the centuries, genius and geometry have helped us gaze beyond our horizons. What can you prove about space? How do you know where you are? Can space be curved? How many dimensions are there? How does geometry explain the natural order and unity of the cosmos? These are the questions behind the five geometric revolutions of world history.

It started with a little scheme hatched by Pythagoras: to employ mathematics as the abstract system of rules that can model the physical universe. Then came a concept of space removed from the ground we trod upon, or the water we swam through. It was the birth of abstraction and proof. Soon the Greeks seemed to be able to find geometric answers to every scientific question, from the theory of the lever to the orbits of the heavenly bodies. But Greek civilization declined and the Romans conquered the Western world. One day just before Easter in A.D. 415, a woman was pulled from a chariot and killed by an ignorant mob. This scholar, devoted to geometry, to Pythagoras, and to rational thought, was the last famous scholar to work in the library at Alexandria before the descent of civilization into the thousand years of the Dark Ages.

Soon after civilization reemerged, so did geometry, but it was a new kind of geometry. It came from a man most civilized — he liked to gamble, sleep until the afternoon, and criticize the Greeks because he considered their method of geometric proof too taxing. To save mental labor, René Descartes married geometry and number. With his idea of coordinates, place and shape could be manipulated as never before, and number could be visualized geometrically. These techniques enabled calculus and the development of modern technology. Thanks to Descartes, geometric concepts such as coordinates and graphs, sines and cosines, vectors and tensors, angles and curvature, appear in every context of physics from solid state electronics to the large-scale structure of space-time, from the technology of transistors and computers to lasers and space travel. But Descartes's work also enabled a more abstract — and revolutionary — idea, the idea of curved space. Do all triangles really have angle sums of 180 degrees, or is that only true if the triangle is on a flat piece of paper? It is not just a question of origami. The mathematics of curved space caused a revolution in the logical foundations, not only of geometry but of all of mathematics. And it made possible Einstein's theory of relativity. Einstein's geometric theory of space and that extra dimension, time, and of the relation of space-time to matter and energy, represented a paradigm change of a magnitude not seen in physics since Newton. It sure seemed radical. But that was nothing, compared to the latest revolution.

One day in June 1984, a scientist announced that he had made a breakthrough in the theory that would explain everything from why subatomic particles exist, and how they interact, to the large-scale structure of space-time and the nature of black holes. This man believed that the key to understanding the unity and order of the universe lies in geometry — geometry of a new and rather bizarre nature. He was carried off the stage by a group of men in white uniforms.

It turned out the event was staged. But the sentiment and genius were real. John Schwarz had been working for a decade and a half on a theory, called string theory, that most physicists reacted to in much the same way they would react to a stranger with a crazed expression asking for money on the street. Today, most physicists believe that string theory is correct: the geometry of space is responsible for the physical laws governing that which exists within it.

The manifesto of the seminal revolution in geometry was written by a mystery man named Euclid. If you don't recall much of that deadly subject called Euclidean Geometry, it is probably because you slept through it. To gaze upon geometry the way it is usually presented is a good way to turn a young mind to stone. But Euclidean geometry is actually an exciting subject, and Euclid's work 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 through which the nature of our universe has been revealed. And as his geometry has passed through four more revolutions, scientists and mathematicians have shattered theologians' beliefs, destroyed philosophers' treasured worldviews, and forced us to reexamine and reimagine our place in the cosmos. These revolutions, and the prophets and stories behind them, are the subject of this book.

Copyright © 2001 by Leonard Mlodinow

Product Details

ISBN:
9780684865249
Author:
Mlodinow, Leonard
Publisher:
Free Press
Location:
New York
Subject:
General
Subject:
General science
Subject:
History
Subject:
Geometry - General
Subject:
Geometry
Subject:
History -- Philosophy.
Subject:
General Mathematics
Subject:
Geometry -- History.
Subject:
Mathematics -- History.
Copyright:
Edition Number:
1st Touchstone ed.
Edition Description:
B102
Series Volume:
967
Publication Date:
April 2002
Binding:
TRADE PAPER
Grade Level:
General/trade
Language:
English
Illustrations:
Yes
Pages:
320
Dimensions:
9.25 x 5.25 in 10.71 oz

Other books you might like

  1. Story of Mathematics Used Trade Paper $5.95
  2. Joy of Mathematics Rev Edition Used Trade Paper $4.95
  3. The Joy of Pi Used Trade Paper $5.95
  4. What Counts: How Every Brain is... Used Hardcover $6.50
  5. The Beginner's Guide to Constructing...
    Used Trade Paper $12.95
  6. Trigonometry New Trade Paper $38.25

Related Subjects


Reference » Science Reference » General
Science and Mathematics » Featured Titles in Tech » General
Science and Mathematics » Mathematics » Calculus » Precalculus
Science and Mathematics » Mathematics » Geometry » Geometry and Trigonometry
Science and Mathematics » Mathematics » History

Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace Used Trade Paper
0 stars - 0 reviews
$8.95 In Stock
Product details 320 pages Free Press - English 9780684865249 Reviews:
spacer
spacer
  • back to top

FOLLOW US ON...

     
Powell's City of Books is an independent bookstore in Portland, Oregon, that fills a whole city block with more than a million new, used, and out of print books. Shop those shelves — plus literally millions more books, DVDs, and gifts — here at Powells.com.