You can't trisect an angle with a compass and straightedge

books:


	• Why Beauty Is Truth: The History of Symmetry Ian Stewart Basic Books, 2007 - 304 pages average customer review:based on 16 reviews view larger image
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highly recommended

Nearly gets the balance of history and math right

This book tries to pull off a difficult trick: being both a history of mathematics and mathematicians, and also a primer on group theory and symmetry. Glossing over the real technical details, Stewart does a good job explaining the math, but a good deal of it still went over my head--although he tries to keep things simple, he expects you to actually *remember* some key parts of high-school math.

Math sections alternate with passages about the lives of the discoverers of various theoretical advances. As much as the math gets simplified, so does the history. Facts, people, and context go whipping by at points, reducing some important information down to single lonely sentences.

And amazingly, for a book titled "Why Beauty is Truth", there's no single clear definition of what (mathematical) "beauty" is. There are plenty of references to "elegant" equations, or even beautiful ones, but no statement about why mathematicians might find them so, even though I think such a definition is quite simple. David Gelernter's wonderful definition from Machine Beauty would be ideal: "simplicity plus power equals beauty." That is, an equation which is simpler, and which gives useful leverage or has predictive abilities, is elegant and beautiful. A long equation tailored to a specific problem is merely functional.

The most compelling idea in the book, which appears a few times, is that the structure of mathematics is not merely an analogy or functional metaphor for "the real world" but is an actual, literal description of it and can even make testable predictions about it. The terrific book Strange Matters: Undiscovered Ideas at the Frontiers of Space and Time looks at that predictive power in more depth, specifically in the field of cosmology.

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I liked it, but, you can trisect an angle

Not only can Achilles catch a tortise, he can also trisect an angle.

It just takes infinite iterations.

As iterations -> infinity, angle -> trisection

I figured it out in eighth grade, and later was glad to see that the theory of limits wasn't something I'd made up.

But for a book that is a combination of light history and fun explorations, it makes for a good holiday read.

Other than repeating the old saw that you can't tri-sect an angle one too many times.

You just have to be very patient. ;)

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"Beauty, Truth & Mathematics via Transformation"

"Why Beauty Is Truth: A History of Symmetry", by Ian Stewart, Basic Books, NY 2007. ISBN-13: 978-0-465-08236-0. HC 290/280 pgs., includes Preface, Further Readings, Index & a few cartoons. 9 1/2" x 6 1/2".

Stewart, Mathematics Professor, Warwick, authored six prior books and here he provides an entertaining survey of the history of symmetry with especial reference to mathematical purity, elegance, simplicity and symmetry of divers sorts, group theory, imaginary numbers & much more.

Tolerantly technical, and despite reader caution not need to complete calculations, those reader's lacking math background and basic comprehension of quantum and particle physics will be awash. The emboldened cover enveiglement, an Azure Lepidoptera, is enticing but not pertinent to book's contents. Written in engaging, but oft meandering prose attitude as met in Mario Livio's "The Golden Ratio", but more earthy than encountered in 'historicisms', tolerably askance or tangential, it is chock full of anecdotal informatories adding to it's intelligibility. String theory, supersymmetry and Feynam sketches are helpful as apt diagrams.

The semi-chronological format delves into contributions made by the "usual suspects" of math and physics -- beginning in ancient Babylon (60 miles South of Baghdad), and onto Euclid, Einstein, Weber, Planck, Witten, etc. An excellent primer and a good read.

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You can't trisect an angle with a compass and straightedge

Why Beauty Is Truth: A History of Symmetry
S. Marsh statement "you can trisect an angle" is not true in its historical context.

Historical context: It is not possible to trisect all angles using only a compass and a straightedge (unmarked ruler).

In his book, Stewart says that it is possible to compute values to great precision,(which includes using iteration) but not by compass and ruler. He does mention that it is possible to trisect some angles, specifically mentioning 180 which trisects to 60 which can be constructed by making a regular hexagon. But trisecting 60 degrees by compass and ruler to produce 20 degrees is impossible, Note that 20 is the exact value of a trisected 60 degree angle but you cannot construct that angle, with a straightedge and compass.

As Stewart makes clear in this book, the important thing is not that you can't, but why you can't. And the why leads to group theory and other advances.

I found this book to be extremely interesting. Group theory is new to me. I found this book is an introduction as to why it was important to Einstein and to modern physics.

I recommend this book.

I found the following on-line tutorial on Galois theory useful:
http://nrich.maths.org/public/viewer.php?obj_id=1422

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reviews: page 1, 2, 3, 4

At the heart of relativity theory, quantum mechanics, string theory, and much of modern cosmology lies one concept: symmetry. In Why Beauty Is Truth, world-famous mathematician Ian Stewart narrates the history of the emergence of this remarkable area of study. Stewart introduces us to such characters as the Renaissance Italian genius, rogue, scholar, and gambler Girolamo Cardano, who stole the modern method of solving cubic equations and published it in the first important book on algebra, and the young revolutionary Evariste Galois, who refashioned the whole of mathematics and founded the field of group theory only to die in a pointless duel over a woman before his work was published. Stewart also explores the strange numerology of real mathematics, in which particular numbers have unique and unpredictable properties related to symmetry. He shows how Wilhelm Killing discovered ?Lie groups? with 14, 52, 78, 133, and 248 dimensions-groups whose very existence is a profound puzzle. Finally, Stewart describes the world beyond superstrings: the ?octonionic? symmetries that may explain the very existence of the universe.

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